If $A=\left[\begin{array}{ccc}1 & 2 & 3 \ -1 & 1 & 2 \ 1 & 2 & 4\end{array} ight]$ and $A_{i j}$ is a cofactor of $a_{i j}$, then the value of $a_{21} A_{21}+a_{22} A_{22}+a_{23} A_{23}$ is

Question

If $A=\left[\begin{array}{ccc}1 & 2 & 3 \ -1 & 1 & 2 \ 1 & 2 & 4\end{array}ight]$ and $A_{i j}$ is a cofactor of $a_{i j}$, then the value of $a_{21} A_{21}+a_{22} A_{22}+a_{23} A_{23}$ is

Aicharya · Accepted Answer

$\begin{aligned} & \mathrm{a}_{21}=-1, \mathrm{a}_{22}=1, \mathrm{a}_{23}=2 \ & \mathrm{~A}_{21}=(-1)^{2+1}\left|\begin{array}{ll}2 & 3 \ 2 & 4\end{array}ight|=(-1)(2)=-2 \ & \mathrm{~A}_{22}=(-1)^{2+2}\left|\begin{array}{ll}1 & 3 \ 1 & 4\end{array}ight|=1(1)=1 \ & \quad \mathrm{~A}_{23}=(-1)^{2+3}\left|\begin{array}{ll}1 & 2 \ 1 & 2\end{array}ight|=(-1)(0)=0 \ & 	herefore \quad \mathrm{a}_{21} \mathrm{~A}_{21}+\mathrm{a}_{22} \mathrm{~A}_{22}+\mathrm{a}_{23} \mathrm{~A}_{23}=(-1)(-2)+1(1)+2(0) \ &=3\end{aligned}$

If $A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$ and $A_{i j}$ is a…

Solution