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Question

Compute the determinants using cofactor expansion along any row or column that seems convenient.

413224110\left| \begin{array} { r r r } { - 4 } & { 1 } & { 3 } \\ { 2 } & { - 2 } & { 4 } \\ { 1 } & { - 1 } & { 0 } \end{array} \right|

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

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

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

det(A)=aj1Cj1+aj2Cj2++ajnCjn\det(A)=a_{j1}C_{j1}+a_{j2}C_{j2}+\dots+a_{jn}C_{jn}

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

det(A)=a1jC1j+a2jC2j++anjCnj\det(A)=a_{1j}C_{1j}+a_{2j}C_{2j}+\dots+a_{nj}C_{nj}

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

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