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Expansion along any row and column

<aside> 💡 $det(A)=\sum_{j=1}^n(-1)^{i+j}a_{ij}M_{ij}$, for a fixed $i$. $=\sum_{i=1}^n(-1)^{i+j}a_{ij}M_{ij}$, for a fixed $j$.

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Important Properties and Identities

Property 1: Determinant of a product is product of the determinants.

        Related Identity: $det(AB)=det(A)det(B)$

Some Results related to Property 1:

• $det(A^n)=det(A)^n$
• $det(A^{-1})=\frac 1 {det(A)} = det(A)^{-1}$
• $det(P^{-1}AP)=det(A)$
• $det(AB)=det(BA)$
• $det(A^TA)=det(A)^2$

Property 2: Switching two rows or columns changes the sign of the determinant.

Property 3: Adding multiples of a row(column) to another row(column) leaves the determinant unchanged.

Property 4: Scalar Multiplication of a row(column) by a constant $t$ multiplies the determinant by $t$.

Note: $det(tA)=t^ndet(A)$, where $t$ is a scalar and $n$ is the order of matrix A. Note: The determinant of a matrix with a row or column of zeroes is 0. Note: The determinant of a matrix in which one row(or column) is a linear combination of other rows(or columns) is zero.