Question

If $A = \begin{bmatrix}5a &-b\\ 3&2\end{bmatrix}$ and $A$ adj $A$ = $AA^T$ , then $5a + b$ is equal to :

• A. -1
• B. 5
• C. 4
• D. 13

Answer 1

Answer: (B)

$A = \begin{bmatrix}5a&-b\\ 3&2\end{bmatrix}$
$A.adj \ A =A.A^{T}$
$ \begin{bmatrix}5a&-b\\ 3&2\end{bmatrix} \begin{bmatrix}2&b\\ -3&5a\end{bmatrix}= \begin{bmatrix}5a&-b\\ 3&2\end{bmatrix}\begin{bmatrix}5a&3\\ -b&2\end{bmatrix}$
$ \begin{bmatrix}10a+3b&0\\ 0&10a+3b\end{bmatrix} = \begin{bmatrix}25a^{2}+b^{2}&15a-2b\\ 15a-2b&13\end{bmatrix} $
Equate, $10 a + 3b = 25 a^2 + b^2$
& $10a + 3b = 13$
& $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 $