91Ó°ÊÓ

The determinant computed by acofactor expansion across the i^th row is

\(\det A = {a_{i1}}{C_{i1}} + {a_{i2}}{C_{i2}} + \cdots + {a_{in}}{C_{in}}\).

Here, A is an \(n \times n\) matrix, and \({C_{ij}} = {\left( { - 1} \right)^{i + j}}{A_{ij}}\).

\(\begin{aligned}{c}\left| {\begin{aligned}{*{20}{c}}2&3&{ - 3}\\4&0&3\\6&1&5\end{aligned}} \right| = {a_{11}}{C_{11}} + {a_{12}}{C_{12}} + {a_{13}}{C_{13}}\\ = {a_{11}}{\left( { - 1} \right)^{1 + 1}}\det {A_{11}} + {a_{12}}{\left( { - 1} \right)^{1 + 2}}\det {A_{12}} + {a_{13}}{\left( { - 1} \right)^{1 + 3}}\det {A_{13}}\\ = 2\left| {\begin{aligned}{*{20}{c}}0&3\\1&5\end{aligned}} \right| - 3\left| {\begin{aligned}{*{20}{c}}4&3\\6&5\end{aligned}} \right| + \left( { - 3} \right)\left| {\begin{aligned}{*{20}{c}}4&0\\6&1\end{aligned}} \right|\\ = 2\left( { - 3} \right) - 3\left( 2 \right) - 3\left( 4 \right)\\ = - 6 - 6 - 12\\ = - 24\end{aligned}\)

Hence, the given determinant using a cofactor expansion across the first row is \( - 24\).