Question

Let A be a square matrix of order 3 such that $\left|A\right|$ = 11 and B be the matrix of confactors of elements of A. Then, ${\left|B\right|}^2$ = ________________.

Answer 1

Given:
A be a square matrix of order 3
$\left|A\right|$ = 11
B be the matrix of cofactors of elements of A

Since, B be the matrix of cofactors of elements of A
$$\begin{gathered} \Rightarrow B={\left(\mathrm{adj}A\right)}^T \\ \Rightarrow \left|B\right|=\left|{\left(\mathrm{adj}A\right)}^T\right| \\ \Rightarrow \left|B\right|=\left|\mathrm{adj}A\right|\left(\because \left|A^T\right|=\left|A\right|\right) \\ \Rightarrow {\left|B\right|}^2={\left|\mathrm{adj}A\right|}^2 \end{gathered}$$

As we know,
$$\begin{gathered} \left|\mathrm{adj}A\right|={\left|A\right|}^{n-1},\mathrm{where}n\mathrm{is}\mathrm{the}\mathrm{order}\mathrm{of}A \\ \Rightarrow \left|B\right|={\left|A\right|}^{n-1} \\ \Rightarrow \left|B\right|={\left|A\right|}^{3-1}\left(\because \mathrm{Order}\mathrm{of}A\mathrm{is}3\right) \\ \Rightarrow \left|B\right|={\left|A\right|}^2 \\ \Rightarrow \left|B\right|={\left(11\right)}^2\left(\because \left|A\right|=11\right) \\ \Rightarrow \left|B\right|=121 \\ \Rightarrow {\left|B\right|}^2={\left(121\right)}^2 \\ \Rightarrow {\left|B\right|}^2=14641 \end{gathered}$$


Hence, ${\left|B\right|}^2$ = 14641.