Determinants and matrix operations:
- $\det(AB) = \det(A)\det(B)$
- $\det(A^T) = \det(A)$
- $\det(cA) = c^n \det(A)$ for $n \times n$ matrix
- $\det(A^{-1}) = 1/\det(A)$
- Row swap changes sign of determinant
- Row scaling by $c$ multiplies determinant by $c$
- Row replacement does not change determinant
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- Definition of determinant of a matrix as a product of the diagonal entries in a non-scaled echelon form.Reduce square $A$ to upper triangular $U$ using row additions and $s$ row interchanges, without scaling rows. Then...
- The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form.Upper triangular matrices have determinant equal to their diagonal product. Row replacement preserves determinant and...
- Definition of the determinant in terms of the effect of elementary row operationsThe determinant is normalized by $\det I=1$ and obeys three row-operation rules: swapping rows multiplies it by $-1$;...
- The permutation expansion for determinantsFor $A\in\mathbb F^{n\times n}$, $\det A=\sum {\sigma\in S n}\operatorname{sgn}(\sigma)\prod {i=1}^n a...
- A matrix and its transpose have the same determinant.For square $A$, $\det A^T=\det A$. In the permutation expansion, relabeling each permutation by its inverse matches...
- Determinants multiply: det(AB) = det(A)det(B)For square matrices of the same size, $\det(AB)=\det A\,\det B$. This multiplicativity explains why a product is...
- The determinant of the inverse of A is the reciprocal of the determinant of A.For invertible $A$, $\det(A^{-1})=1/\det A$. Apply multiplicativity to $AA^{-1}=I$ to obtain $\det A\det(A^{-1})=1$....
- The determinant of a block diagonal matrix is the product of the determinants of the blocks.For a block diagonal matrix $\operatorname{diag}(B 1,\ldots,B k)$ with square blocks, the determinant is $\prod i\det...
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