Question

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $A\left(\text{adj}A\right)=KI$, then the value of $K$ is:

• A. $2$
• B. $-2$
• C. $10$
• D. $-10$

Answer 1

Answer: (B)

$-2$

Given, $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$
Adjoint of $A=\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]$
$\therefore$ $A\left(\text{adj}A\right)=KI$
$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}4& 2\\ -3& 1\end{array}\right]=K\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right]=\left[\begin{array}{cc}K& 0\\ 0& K\end{array}\right]$
$\therefore$ $K=-2$