Question

For any $2\times 2$ matrix $A$, if $A[\text{adj}.A]=\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$ ,then $\left|A\right|$ equals

• A. $0$
• B. $10$
• C. $20$
• D. $100$

Answer 1

The correct option is B $10$

$A\left[AdjA\right]=\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$

$A\left(AdjA\right)=10I-\left(1\right)$

$A^{-1}=\frac{AdjA}{\left(A\right)}$

$\frac{I}{A}=\frac{AdjA}{\left(A\right)}$

$\left(A\right)I=A.AdjA-\left(2\right)$

Comparing $\left(1\right)$ and $\left(2\right)$

$\Rightarrow \left(A\right)=10$

Option-B