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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement

11.2.8  Submatrices

Syntaxcsubma 20201 Syntax for submatrices of a square matrix.
class subMat

Definitiondf-subma 20202* Define the submatrices of a square matrix. A submatrix is obtained by deleting a row and a column of the original matrix. Since the indices of a matrix need not to be sequential integers, it does not matter that there may be gaps in the numbering of the indices for the submatrix. The determinants of such submatrices are called the "minors" of the original matrix. (Contributed by AV, 27-Dec-2018.)
subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))

Theoremsubmabas 20203* Any subset of the index set of a square matrix defines a submatrix of the matrix. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵𝐷𝑁) → (𝑖𝐷, 𝑗𝐷 ↦ (𝑖𝑀𝑗)) ∈ (Base‘(𝐷 Mat 𝑅)))

Theoremsubmafval 20204* First substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))

Theoremsubmaval0 20205* Second substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))

Theoremsubmaval 20206* Third substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))

Theoremsubmaeval 20207 An entry of a submatrix of a square matrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼 ∈ (𝑁 ∖ {𝐾}) ∧ 𝐽 ∈ (𝑁 ∖ {𝐿}))) → (𝐼(𝐾(𝑄𝑀)𝐿)𝐽) = (𝐼𝑀𝐽))

Theorem1marepvsma1 20208 The submatrix of the identity matrix with the ith column replaced by the vector obtained by removing the ith row and the ith column is an identity matrix. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)
𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)    &    1 = (1r‘(𝑁 Mat 𝑅))    &   𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)       (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝐼((𝑁 subMat 𝑅)‘𝑋)𝐼) = (1r‘((𝑁 ∖ {𝐼}) Mat 𝑅)))

11.3  The determinant

11.3.1  Definition and basic properties

Syntaxcmdat 20209 Syntax for the matrix determinant function.

Definitiondf-mdet 20210* Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 20212). The properties of the axiomatic definition of a determinant according to [Weierstrass] p. 272 are derived from this definition as theorems: "The determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring". The functionality is shown by mdetf 20220. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 20227, the homogeneity by mdetrsca 20228. Furthermore, it is shown that the determinant function is alternating (see mdetralt 20233) and normalized (see mdet1 20226). Finally, the uniqueness is shown by mdetuni 20247. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 20212. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 10-Jul-2018.)
maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))))

Theoremmdetfval 20211* First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 9-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))

Theoremmdetleib 20212* Full substitution of our determinant definition (also known as Leibniz' Formula, expanding by columns). Proposition 4.6 in [Lang] p. 514. (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by SO, 9-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))

Theoremmdetleib2 20213* Leibniz' formula can also be expanded by rows. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ (𝑥𝑀(𝑝𝑥))))))))

Theoremnfimdetndef 20214 The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)       (𝑁 ∉ Fin → 𝐷 = ∅)

Theoremmdetfval1 20215* First substitution of an alternative determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 27-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ ((𝑌‘(𝑆𝑝)) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))

Theoremmdetleib1 20216* Full substitution of an alternative determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by AV, 26-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ ((𝑌‘(𝑆𝑝)) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))

Theoremmdet0pr 20217 The determinant for 0-dimensional matrices is a singleton containing an ordered pair with the singleton containing the empty set as first component, and the singleton containing the 1 element of the underlying ring as second component. (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → (∅ maDet 𝑅) = {⟨∅, (1r𝑅)⟩})

Theoremmdet0f1o 20218 The determinant for 0-dimensional matrices is a one-to-one function from the singleton containing the empty set onto the singleton containing the 1 element of the underlying ring.function x is . (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → (∅ maDet 𝑅):{∅}–1-1-onto→{(1r𝑅)})

Theoremmdet0fv0 20219 The determinant of a 0-dimensional matrix is the 1 element of the underlying ring . (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → ((∅ maDet 𝑅)‘∅) = (1r𝑅))

Theoremmdetf 20220 Functionality of the determinant, see also definition in [Lang] p. 513. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)       (𝑅 ∈ CRing → 𝐷:𝐵𝐾)

Theoremmdetcl 20221 The determinant evaluates to an element of the base ring. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 7-Feb-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐷𝑀) ∈ 𝐾)

Theoremm1detdiag 20222 The determinant of a 1-dimensional matrix equals its (single) entry. (Contributed by AV, 6-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼𝑉) ∧ 𝑀𝐵) → (𝐷𝑀) = (𝐼𝑀𝐼))

Theoremmdetdiaglem 20223* Lemma for mdetdiag 20224. Previously part of proof for mdet1 20226. (Contributed by SO, 10-Jul-2018.) (Revised by AV, 17-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)    &    0 = (0g𝑅)    &   𝐻 = (Base‘(SymGrp‘𝑁))    &   𝑍 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀𝐵) ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ) ∧ (𝑃𝐻𝑃 ≠ ( I ↾ 𝑁))) → (((𝑍𝑆)‘𝑃) · (𝐺 Σg (𝑘𝑁 ↦ ((𝑃𝑘)𝑀𝑘)))) = 0 )

Theoremmdetdiag 20224* The determinant of a diagonal matrix is the product of the entries in the diagonal. (Contributed by AV, 17-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀𝐵) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ) → (𝐷𝑀) = (𝐺 Σg (𝑘𝑁 ↦ (𝑘𝑀𝑘)))))

Theoremmdetdiagid 20225* The determinant of a diagonal matrix with identical entries is the power of the entry in the diagonal. (Contributed by AV, 17-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)    &    0 = (0g𝑅)    &   𝐶 = (Base‘𝑅)    &    · = (.g𝐺)       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀𝐵𝑋𝐶)) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑋, 0 ) → (𝐷𝑀) = ((#‘𝑁) · 𝑋)))

Theoremmdet1 20226 The determinant of the identity matrix is 1, i.e. the determinant function is normalized, see also definition in [Lang] p. 513. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 25-Nov-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐼 = (1r𝐴)    &    1 = (1r𝑅)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐷𝐼) = 1 )

Theoremmdetrlin 20227 The determinant function is additive for each row: The matrices X, Y, Z are identical except for the I's row, and the I's row of the matrix X is the componentwise sum of the I's row of the matrices Y and Z. In this case the determinant of X is the sum of the determinants of Y and Z. (Contributed by SO, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    + = (+g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐼𝑁)    &   (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((𝑌 ↾ ({𝐼} × 𝑁)) ∘𝑓 + (𝑍 ↾ ({𝐼} × 𝑁))))    &   (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑌 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))    &   (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))       (𝜑 → (𝐷𝑋) = ((𝐷𝑌) + (𝐷𝑍)))

Theoremmdetrsca 20228 The determinant function is homogeneous for each row: The matrices X and Z are identical except for the I's row, and the I's row of the matrix X is the componentwise product of the I's row of the matrix Z and the scalar Y. In this case the determinant of X is the determinant of Z multiplied by Y. (Contributed by SO, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐾)    &   (𝜑𝑍𝐵)    &   (𝜑𝐼𝑁)    &   (𝜑 → (𝑋 ↾ ({𝐼} × 𝑁)) = ((({𝐼} × 𝑁) × {𝑌}) ∘𝑓 · (𝑍 ↾ ({𝐼} × 𝑁))))    &   (𝜑 → (𝑋 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)) = (𝑍 ↾ ((𝑁 ∖ {𝐼}) × 𝑁)))       (𝜑 → (𝐷𝑋) = (𝑌 · (𝐷𝑍)))

Theoremmdetrsca2 20229* The determinant function is homogeneous for each row (matrices are given explicitly by their entries). (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐾 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑋𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑌𝐾)    &   (𝜑𝐹𝐾)    &   (𝜑𝐼𝑁)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, (𝐹 · 𝑋), 𝑌))) = (𝐹 · (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑌)))))

Theoremmdetr0 20230* The determinant of a matrix with a row containing only 0's is 0. (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑋𝐾)    &   (𝜑𝐼𝑁)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 )

Theoremmdet0 20231 The determinant of the zero matrix (of dimension greater 0!) is 0. (Contributed by AV, 17-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑍 = (0g𝐴)    &    0 = (0g𝑅)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) → (𝐷𝑍) = 0 )

Theoremmdetrlin2 20232* The determinant function is additive for each row (matrices are given explicitly by their entries). (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑋𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑌𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑍𝐾)    &   (𝜑𝐼𝑁)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + 𝑌), 𝑍))) = ((𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍))) + (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑌, 𝑍)))))

Theoremmdetralt 20233* The determinant function is alternating regarding rows: if a matrix has two identical rows, its determinant is 0. Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 23-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝐼𝐽)    &   (𝜑 → ∀𝑎𝑁 (𝐼𝑋𝑎) = (𝐽𝑋𝑎))       (𝜑 → (𝐷𝑋) = 0 )

Theoremmdetralt2 20234* The determinant function is alternating regarding rows (matrix is given explicitly by its entries). (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑗𝑁) → 𝑋𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑌𝐾)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝐼𝐽)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))) = 0 )

Theoremmdetero 20235* The determinant function is multilinear (additive and homogeneous for each row (matrices are given explicitly by their entries). Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑗𝑁) → 𝑋𝐾)    &   ((𝜑𝑗𝑁) → 𝑌𝐾)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝑍𝐾)    &   (𝜑𝑊𝐾)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝐼𝐽)       (𝜑 → (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝐷‘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))

Theoremmdettpos 20236 Determinant is invariant under transposition. Proposition 4.8 in [Lang] p. 514. (Contributed by Stefan O'Rear, 9-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐷‘tpos 𝑀) = (𝐷𝑀))

Theoremmdetunilem1 20237* Lemma for mdetuni 20247. (Contributed by SO, 14-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))       (((𝜑𝐸𝐵 ∧ ∀𝑤𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹𝑁𝐺𝑁𝐹𝐺)) → (𝐷𝐸) = 0 )

Theoremmdetunilem2 20238* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜓𝜑)    &   (𝜓 → (𝐸𝑁𝐺𝑁𝐸𝐺))    &   ((𝜓𝑏𝑁) → 𝐹𝐾)    &   ((𝜓𝑎𝑁𝑏𝑁) → 𝐻𝐾)       (𝜓 → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))) = 0 )

Theoremmdetunilem3 20239* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))       (((𝜑𝐸𝐵𝐹𝐵) ∧ (𝐺𝐵𝐻𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((𝐹 ↾ ({𝐻} × 𝑁)) ∘𝑓 + (𝐺 ↾ ({𝐻} × 𝑁)))) ∧ ((𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐹 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷𝐸) = ((𝐷𝐹) + (𝐷𝐺)))

Theoremmdetunilem4 20240* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))       ((𝜑 ∧ (𝐸𝐵𝐹𝐾𝐺𝐵) ∧ (𝐻𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘𝑓 · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷𝐸) = (𝐹 · (𝐷𝐺)))

Theoremmdetunilem5 20241* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜓𝜑)    &   (𝜓𝐸𝑁)    &   ((𝜓𝑎𝑁𝑏𝑁) → (𝐹𝐾𝐺𝐾𝐻𝐾))       (𝜓 → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) = ((𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) + (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)))))

Theoremmdetunilem6 20242* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜓𝜑)    &   (𝜓 → (𝐸𝑁𝐹𝑁𝐸𝐹))    &   ((𝜓𝑏𝑁) → (𝐺𝐾𝐻𝐾))    &   ((𝜓𝑎𝑁𝑏𝑁) → 𝐼𝐾)       (𝜓 → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) = ((invg𝑅)‘(𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))))

Theoremmdetunilem7 20243* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))       ((𝜑𝐸:𝑁1-1-onto𝑁𝐹𝐵) → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ ((𝐸𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷𝐹)))

Theoremmdetunilem8 20244* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜑 → (𝐷‘(1r𝐴)) = 0 )       ((𝜑𝐸:𝑁𝑁) → (𝐷‘(𝑎𝑁, 𝑏𝑁 ↦ if((𝐸𝑎) = 𝑏, 1 , 0 ))) = 0 )

Theoremmdetunilem9 20245* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   (𝜑 → (𝐷‘(1r𝐴)) = 0 )    &   𝑌 = {𝑥 ∣ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )}       (𝜑𝐷 = (𝐵 × { 0 }))

Theoremmdetuni0 20246* Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   𝐸 = (𝑁 maDet 𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐷𝐹) = ((𝐷‘(1r𝐴)) · (𝐸𝐹)))

Theoremmdetuni 20247* According to the definition in [Weierstrass] p. 272, the determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring. So for any multilinear (mdetuni.li and mdetuni.sc), alternating (mdetuni.al) and normalized (mdetuni.no) function D (mdetuni.ff) from the algebra of square matrices (mdetuni.a) to their underlying commutative ring (mdetuni.cr), the function value of this function D for a matrix F (mdetuni.f) is the determinant of this matrix. (Contributed by Stefan O'Rear, 15-Jul-2018.) (Revised by Alexander van der Vekens, 8-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐷:𝐵𝐾)    &   (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))    &   (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))    &   (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))    &   𝐸 = (𝑁 maDet 𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐹𝐵)    &   (𝜑 → (𝐷‘(1r𝐴)) = 1 )       (𝜑 → (𝐷𝐹) = (𝐸𝐹))

Theoremmdetmul 20248 Multiplicativity of the determinant function: the determinant of a matrix product of square matrices equals the product of their determinants. Proposition 4.15 in [Lang] p. 517. (Contributed by Stefan O'Rear, 16-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐷 = (𝑁 maDet 𝑅)    &    · = (.r𝑅)    &    = (.r𝐴)       ((𝑅 ∈ CRing ∧ 𝐹𝐵𝐺𝐵) → (𝐷‘(𝐹 𝐺)) = ((𝐷𝐹) · (𝐷𝐺)))

11.3.2  Determinants of 2 x 2 -matrices

Theoremm2detleiblem1 20249 Lemma 1 for m2detleib 20256. (Contributed by AV, 12-Dec-2018.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑄𝑃) → (𝑌‘(𝑆𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g𝑅) 1 ))

Theoremm2detleiblem5 20250 Lemma 5 for m2detleib 20256. (Contributed by AV, 20-Dec-2018.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → (𝑌‘(𝑆𝑄)) = 1 )

Theoremm2detleiblem6 20251 Lemma 6 for m2detleib 20256. (Contributed by AV, 20-Dec-2018.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    1 = (1r𝑅)    &   𝐼 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩}) → (𝑌‘(𝑆𝑄)) = (𝐼1 ))

Theoremm2detleiblem7 20252 Lemma 7 for m2detleib 20256. (Contributed by AV, 20-Dec-2018.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    1 = (1r𝑅)    &   𝐼 = (invg𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ 𝑍 ∈ (Base‘𝑅)) → (𝑋(+g𝑅)((𝐼1 ) · 𝑍)) = (𝑋 𝑍))

Theoremm2detleiblem2 20253* Lemma 2 for m2detleib 20256. (Contributed by AV, 16-Dec-2018.) (Proof shortened by AV, 1-Jan-2019.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵) → (𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛))) ∈ (Base‘𝑅))

Theoremm2detleiblem3 20254* Lemma 3 for m2detleib 20256. (Contributed by AV, 16-Dec-2018.) (Proof shortened by AV, 2-Jan-2019.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)    &    · = (+g𝐺)       ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀𝐵) → (𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))

Theoremm2detleiblem4 20255* Lemma 4 for m2detleib 20256. (Contributed by AV, 20-Dec-2018.) (Proof shortened by AV, 2-Jan-2019.)
𝑁 = {1, 2}    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)    &    · = (+g𝐺)       ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀𝐵) → (𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))

Theoremm2detleib 20256 Leibniz' Formula for 2x2-matrices. (Contributed by AV, 21-Dec-2018.) (Revised by AV, 26-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝑁 = {1, 2}    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (-g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝐷𝑀) = (((1𝑀1) · (2𝑀2)) ((2𝑀1) · (1𝑀2))))

Syntaxcminmar1 20258 Syntax for the minor matrices of a square matrix.
class minMatR1

Definitiondf-madu 20259* Define the adjugate or adjunct (matrix of cofactors) of a square matrix. This definition gives the standard cofactors, however the internal minors are not the standard minors, see definition in [Lang] p. 518. (Contributed by Stefan O'Rear, 7-Sep-2015.) (Revised by SO, 10-Jul-2018.)
maAdju = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖𝑛, 𝑗𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘𝑛, 𝑙𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))))))

Definitiondf-minmar1 20260* Define the matrices whose determinants are the minors of a square matrix. In contrast to the standard definition of minors, a row is replaced by 0's and one 1 instead of deleting the column and row (e.g., definition in [Lang] p. 515). By this, the determinant of such a matrix is equal to the minor determined in the standard way (as determinant of a submatrix, see df-subma 20202- note that the matrix is transposed compared with the submatrix defined in df-subma 20202, but this does not matter because the determinants are the same, see mdettpos 20236). Such matrices are used in the definition of an adjunct of a square matrix, see df-madu 20259. (Contributed by AV, 27-Dec-2018.)
minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))))

Theoremmndifsplit 20261 Lemma for maducoeval2 20265. (Contributed by SO, 16-Jul-2018.)
𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))

Theoremmadufval 20262* First substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       𝐽 = (𝑚𝐵 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))

Theoremmaduval 20263* Second substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑀𝐵 → (𝐽𝑀) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))

Theoremmaducoeval 20264* An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))

Theoremmaducoeval2 20265* An entry of the adjunct (cofactor) matrix. (Contributed by SO, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))

Theoremmaduf 20266 Creating the adjunct of matrices is a function from the set of matrices into the set of matrices. (Contributed by Stefan O'Rear, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑅 ∈ CRing → 𝐽:𝐵𝐵)

Theoremmadutpos 20267 The adjuct of a transposed matrix is the transposition of the adjunct of the matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽𝑀))

Theoremmadugsum 20268* The determinant of a matrix with a row 𝐿 consisting of the same element 𝑋 is the sum of the elements of the 𝐿-th column of the adjunct of the matrix multiplied with 𝑋. (Contributed by Stefan O'Rear, 16-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐷 = (𝑁 maDet 𝑅)    &    · = (.r𝑅)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝑀𝐵)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑖𝑁) → 𝑋𝐾)    &   (𝜑𝐿𝑁)       (𝜑 → (𝑅 Σg (𝑖𝑁 ↦ (𝑋 · (𝑖(𝐽𝑀)𝐿)))) = (𝐷‘(𝑗𝑁, 𝑖𝑁 ↦ if(𝑗 = 𝐿, 𝑋, (𝑗𝑀𝑖)))))

Theoremmadurid 20269 Multiplying a matrix with its adjunct results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 16-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐷 = (𝑁 maDet 𝑅)    &    1 = (1r𝐴)    &    · = (.r𝐴)    &    = ( ·𝑠𝐴)       ((𝑀𝐵𝑅 ∈ CRing) → (𝑀 · (𝐽𝑀)) = ((𝐷𝑀) 1 ))

Theoremmadulid 20270 Multiplying the adjunct of a matrix with the matrix results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐷 = (𝑁 maDet 𝑅)    &    1 = (1r𝐴)    &    · = (.r𝐴)    &    = ( ·𝑠𝐴)       ((𝑀𝐵𝑅 ∈ CRing) → ((𝐽𝑀) · 𝑀) = ((𝐷𝑀) 1 ))

Theoremminmar1fval 20271* First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 minMatR1 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))

Theoremminmar1val0 20272* Second substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 minMatR1 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))))

Theoremminmar1val 20273* Third substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 minMatR1 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))

Theoremminmar1eval 20274 An entry of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 minMatR1 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑄𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))

Theoremminmar1marrep 20275 The minor matrix is a special case of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅) 1 ))

Theoremminmar1cl 20276 Closure of the row replacement function for square matrices: The matrix for a minor is a matrix. (Contributed by AV, 13-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐿) ∈ 𝐵)

Theoremmaducoevalmin1 20277 The coefficients of an adjunct (matrix of cofactors) expressed as determinants of the minor matrices (alternative definition) of the original matrix. (Contributed by AV, 31-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)       ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))

11.3.4  Laplace expansion of determinants (special case)

According to Wikipedia ("Laplace expansion", 08-Mar-2019, https://en.wikipedia.org/wiki/Laplace_expansion) "In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant det(B) of an n x n -matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n-1) x (n-1)". The expansion is usually performed for a row of matrix B (alternatively for a column of matrix B). The mentioned "sub-matrices" are the matrices resultung from deleting the i-th row and the j-th column of matrix B. The mentioned "weights" (factors/coefficients) are the elements at position i and j in matrix B. If the expansion is performed for a row, the coefficients are the elements of the selected row.

In the following, only the case where the row for the expansion contains only the zero element of the underlying ring except at the diagonal position. By this, the sum for the Laplace expansion is reduced to one summand, consisting of the element at the diagonal position multiplied with the determinant of the corresponding submatrix, see smadiadetg 20298 or smadiadetr 20300.

Theoremsymgmatr01lem 20278* Lemma for symgmatr01 20279. (Contributed by AV, 3-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))       ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄𝑘))) = 𝐵))

Theoremsymgmatr01 20279* Applying a permutation that does not fix a certain element of a set to a second element to an index of a matrix a row with 0's and a 1. (Contributed by AV, 3-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 ))

Theoremgsummatr01lem1 20280* Lemma A for gsummatr01 20284. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}       ((𝑄𝑅𝑋𝑁) → (𝑄𝑋) ∈ 𝑁)

Theoremgsummatr01lem2 20281* Lemma B for gsummatr01 20284. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}       ((𝑄𝑅𝑋𝑁) → (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ (Base‘𝐺) → (𝑋𝐴(𝑄𝑋)) ∈ (Base‘𝐺)))

Theoremgsummatr01lem3 20282* Lemma 1 for gsummatr01 20284. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)))) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛))))(+g𝐺)(𝐾(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝐾))))

Theoremgsummatr01lem4 20283* Lemma 2 for gsummatr01 20284. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       ((((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) ∧ 𝑛 ∈ (𝑁 ∖ {𝐾})) → (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)) = (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄𝑛)))

𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) → (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)))) = (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄𝑛)))))

Theoremmarep01ma 20285* Replacing a row of a square matrix by a row with 0's and a 1 results in a square matrix of the same dimension. (Contributed by AV, 30-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑀𝐵 → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))) ∈ 𝐵)

Theoremsmadiadetlem0 20286* Lemma 0 for smadiadet 20295: The products of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑛)))) = 0 ))

Theoremsmadiadetlem1 20287* Lemma 1 for smadiadet 20295: A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       (((𝑀𝐵𝐾𝑁) ∧ 𝑝𝑃) → (((𝑌𝑆)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))) ∈ (Base‘𝑅))

Theoremsmadiadetlem1a 20288* Lemma 1a for smadiadet 20295: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↦ (((𝑌𝑆)‘𝑝) · (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))))) = 0 )

Theoremsmadiadetlem2 20289* Lemma 2 for smadiadet 20295: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to itself. (Contributed by AV, 31-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ↦ (((𝑌𝑆)‘𝑝) · (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))))) = 0 )

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       (((𝑀𝐵𝐾𝑁) ∧ 𝑄𝑊) → (((𝑌𝑍)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑄𝑛))))) ∈ (Base‘𝑅))

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛)))))):𝑊⟶(Base‘𝑅))

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → ran (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛)))))) ⊆ ((Cntz‘𝑅)‘ran (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛))))))))

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → (𝑅 Σg (𝑝 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} ↦ (((𝑌𝑆)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛))))))) = (𝑅 Σg (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛))))))))

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → (𝑅 Σg (𝑝 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} ↦ (((𝑌𝑆)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))))) = (𝑅 Σg (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛))))))))

Theoremsmadiadet 20295 The determinant of a submatrix of a square matrix obtained by removing a row and a column at the same index equals the determinant of the original matrix with the row replaced with 0's and a 1 at the diagonal position. (Contributed by AV, 31-Jan-2019.) (Proof shortened by AV, 24-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)       ((𝑀𝐵𝐾𝑁) → (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)) = (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾)))

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)       ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))

𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘𝑓 · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))

Theoremsmadiadetg 20298 The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. (Contributed by AV, 14-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆 · (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))

Theoremsmadiadetg0 20299 Lemma for smadiadetr 20300: version of smadiadetg 20298 with all hypotheses defining class variables removed, i.e. all class variables defined in the hypotheses replaced in the theorem by their definition. (Contributed by AV, 15-Feb-2019.)
𝑅 ∈ CRing       ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))

Theoremsmadiadetr 20300 The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. Closed form of smadiadetg 20298. Special case of the "Laplace expansion", see definition in [Lang] p. 515. (Contributed by AV, 15-Feb-2019.)
(((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝑁 Mat 𝑅))) ∧ (𝐾𝑁𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))

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