Question

If $A$ is a $2\times 2$ matrix and $|A|=2,$ then the matrix represented by $A(adjA)$ is equal to

• A. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$
• C. $\left[\begin{array}{cc}1/2& 0\\ 0& 1/2\end{array}\right]$
• D. $\left[\begin{array}{cc}0& 2\\ 2& 0\end{array}\right]$

Answer 1

Answer: (B)

Consider a $2\times 2$ matrix whose
$|A|=2$ $A=\left[\frac{4}{2}\frac{1}{1}\right]$
Now, $adjA=\left[\begin{array}{cc}1& -2\\ -1& 4\end{array}\right]=\left[\begin{array}{cc}1& -1\\ -2& 4\end{array}\right]$
$\Rightarrow$ $A(adjA)=\left[\begin{array}{cc}4& 1\\ 2& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ -2& 4\end{array}\right]$
$=\left[\begin{array}{cc}4-2& -4+4\\ 2-2& -2+4\end{array}\right]$
$=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$