Question

Let $\text{adj}\left(X\right)$ denote the adjoint of matrix $X.$ If $A$ is a non-singular matrix of order $2$ such that $|A+|A|\text{adj}\left(A\right)|=0,$ then the value of $|A-|A|\text{adj}\left(A\right)|$ is

• A. $8$
• B. $4$
• C. $16$
• D. $12$

Answer 1

The correct option is B $4$

Let $A=\left[\begin{array}{cc}{a}_1& a_2\\ a_3& a_4\end{array}\right]$
$\Rightarrow |A|=a_1{a}_4-a_2{a}_3=k$ (say)

$\text{adj}\left(A\right)=\left[\begin{array}{cc}{a}_4& -a_2\\ -a_3& a_1\end{array}\right]$

$|A+|A|\text{adj}\left(A\right)|=0$
$\Rightarrow \left|\begin{array}{cc}{a}_1+k{a}_4& a_2-k{a}_2\\ a_3-k{a}_3& a_4+k{a}_1\end{array}\right|=0$

$\Rightarrow k(1+a_4^2+a_1^2+k^2+2a_2{a}_3)=0$
$\Rightarrow (a_4+a_1{)}^2+1+k^2-2(a_1{a}_4-a_2{a}_3)=0\Rightarrow (a_4+a_1{)}^2+1+k^2-2k=0$
$\Rightarrow (a_4+a_1{)}^2+(k-1{)}^2=0$
$\Rightarrow k=|A|=1$ and $a_4+a_1=0$

$|A-|A|\text{adj}\left(A\right)|=|A-\text{adj}\left(A\right)|(\because |A|=1)$
$=\left[\begin{array}{cc}{a}_1-a_4& 2a_2\\ 2a_3& a_4-a_1\end{array}\right]$

$=2a_1{a}_4-a_1^2-a_4^2-4a_2{a}_3$
$=4a_1{a}_4-(2a_1{a}_4+a_1^2+a_4^2)-4a_2{a}_3$
$=4(a_1{a}_4-a_2{a}_3)-(a_1+a_4{)}^2$
$=4-0$
$=4$