If $\mathrm{A}_{\mathrm{ij}}$ is the cofactor of the element $\mathrm{a}_{\mathrm{ij}}$ of $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ then write the value of $\left(\mathrm{a}_{32} \mathrm{~A}_{32}\right)$.

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If $\mathrm{A}_{\mathrm{ij}}$ is the cofactor of the element $\mathrm{a}_{\mathrm{ij}}$ of $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ then write the value of $\left(\mathrm{a}_{32} \mathrm{~A}_{32}\right)$ .

If $\mathrm{A}_{\mathrm{ij}}$ is the cofactor of the element $\mathrm{a}_{\mathrm{ij}}$ of $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ then write the value of $\left(\mathrm{a}_{32} \mathrm{~A}_{32}\right)$ .

Solution:

Theorem: $A_{i j}$ is found by deleting $j^{t h}$ rowand $j^{t h}$ column, the determinant of left matrix is called cofactor with multiplied by $(-1)^{(i+j)}$
Given: $\mathrm{j}=3$ and $\mathrm{j}=2$ .
$\begin{array}{l} A_{32}=(-1)^{(3+2)}(2 \times 4-6 \times 5) \\ =-1 \times(-22) \end{array}$
$\begin{array}{l} =22 \\ a_{32}=5 \\ a_{32} A_{32}=5 \times 22 \\ =110 \end{array}$