Question

If $A=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]$ and $A$ adj $A$ = $A{A}^T$ , then $5a+b$ is equal to :

• A. -1
• B. 5
• C. 4
• D. 13

Answer 1

Answer: (B)

$A=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]$
$A.adjA=A.A^T$
$\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]\left[\begin{array}{cc}2& b\\ -3& 5a\end{array}\right]=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]\left[\begin{array}{cc}5a& 3\\ -b& 2\end{array}\right]$
$\left[\begin{array}{cc}10a+3b& 0\\ 0& 10a+3b\end{array}\right]=\left[\begin{array}{cc}25a^2+b^2& 15a-2b\\ 15a-2b& 13\end{array}\right]$
Equate, $10a+3b=25a^2+b^2$
& $10a+3b=13$
& $15a-2b=0$
$\frac{a}{2}=\frac{b}{15}=k(let)$
Solving $a=\frac{2}{5},b=3$
So, $5a+b=5\times \frac{2}{5}+3=5$