Question

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

• A. 5
• B. 4
• C. 13
• D. -1

Answer 1

The correct option is A 5

If $A=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]$ and $A\left(adjA\right)=A{A}^T$

To calculate $adjA$ for a matrix :

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$A_C=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ (Cofactors)

$A_C^T=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Similarly,

$adjA=\left[\begin{array}{cc}2& b\\ -3& 5a\end{array}\right]$ and

$A^T=\left[\begin{array}{cc}5a& 3\\ -b& 2\end{array}\right]$

Given in question: $AadjA=A{A}^T$

$adjA=A^T$

Equating the two matrix,

$\left[\begin{array}{cc}2& b\\ -3& 5a\end{array}\right]=\left[\begin{array}{cc}5a& 3\\ -b& 2\end{array}\right]$

$5a=2$

$b=3$

Now,

$5a+b=2+3=5$

Option $\text{A}$ is correct