Question

$A$ be a square matrix of order 2 with $|A|\ne 0$ such that $|A+|A|$ adj $(A)|=0,$ where adj $(A)$ is a adjoint of matrix $A$ then the value of $A-|A|$ adj $(A)\mid$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

Let $A=\left[\begin{array}{cc}m& n\\ p& q\end{array}\right]$, adj $(a)=\left[\begin{array}{cc}p& -n\\ -p& m\end{array}\right]$
Let $|A|=d=mq-np$
$|A+d\mathrm{adj}A|=\left|\begin{array}{cc}m+qd& n\left(1-d\right)\\ p\left(1-d\right)& q+md\end{array}\right|=0$
$\Rightarrow mq+m^{2}d+q^{2}d+mq{d}^2-np+2npd-np{d}^2=0$
$\Rightarrow (mq-np)+(mq-np)d^2+m^{2}d+q^{2}d+2mqd-2d^2=0$
$\Rightarrow \left(d+d^3-2d^2\right)+d\left(m^2+q^2+2mq\right)=0$
$\Rightarrow d\left[(d-1{)}^2+(m+q{)}^2\right]=0\Rightarrow d=1,m+q=0$
Now, $\mid A-d$ adj $|A|=-(m+q{)}^2+4(mq-np)=4d=4$