Question

If $A=\left[a_{ij}\right]$ is a matrix of order $2\times 2$, such that |A| = - 15 and $C_{ij}$ represents the cofactor of $a_{ij}$, then find $a_{21}{C}_{21}+a_{22}{C}_{22}.$

Answer 1

As $A=[a_{ij}{]}_{2\times 2}=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right].$

Consider $|A|=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=a_{21}{C}_{21}+a_{22}{C}_{22}$ (if we expand along $R_2$)
$\therefore |A|=a_{21}{C}_{21}+a_{22}{C}_{22}=-15$