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Question

# Compute the determinants using cofactor expansion along any row or column that seems convenient.$\left| \begin{array} { r r r } { - 4 } & { 1 } & { 3 } \\ { 2 } & { - 2 } & { 4 } \\ { 1 } & { - 1 } & { 0 } \end{array} \right|$

Solutions

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Let $A=[a_{ij}]$ be an $n\times n$ matrix.

We can find the determinant of $A$ either by expanding along a row or by expanding along a column.

1. If we decide to expand along the $j^\text{th}$ row, then the following is the formula.

$\det(A)=a_{j1}C_{j1}+a_{j2}C_{j2}+\dots+a_{jn}C_{jn}$

1. If we decide to expand along the $j^\text{th}$ column, then the following is the formula.

$\det(A)=a_{1j}C_{1j}+a_{2j}C_{2j}+\dots+a_{nj}C_{nj}$

Here, $C_{ij}$ denotes the $(i,j)^\text{th}$ cofactor.

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