Question

65) 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 :
(1) -1
(2) 5
(3) 4
(4) 13

Answer 1

Dear student
$$\begin{gathered} \mathrm{A}=\left[\begin{array}{cc}5\mathrm{a}& -\mathrm{b}\\ 3& 2\end{array}\right] \\ {\mathrm{A}}^{\mathrm{T}}=\left[\begin{array}{cc}5\mathrm{a}& 3\\ -\mathrm{b}& 2\end{array}\right] \\ \mathrm{adj}\mathrm{A}=\left[\begin{array}{cc}2& \mathrm{b}\\ -3& 5\mathrm{a}\end{array}\right] \\ \mathrm{Aadj}\mathrm{A}={\mathrm{AA}}^{\mathrm{T}} \\ \mathrm{Premultipying}\mathrm{by}{\mathrm{A}}^{-1} \\ \Rightarrow \mathrm{adjA}={\mathrm{A}}^{\mathrm{T}} \\ \Rightarrow \left[\begin{array}{cc}2& \mathrm{b}\\ -3& 5\mathrm{a}\end{array}\right]=\left[\begin{array}{cc}5\mathrm{a}& 3\\ -\mathrm{b}& 2\end{array}\right] \\ \Rightarrow 5\mathrm{a}=2,\mathrm{b}=3,5\mathrm{a}=2 \\ \mathbf{\Rightarrow }\mathbf{5}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{=}\mathbf{2}\mathbf{+}\mathbf{3} \\ \mathbf{\Rightarrow }\mathbf{5}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{=}\mathbf{5} \end{gathered}$$
Regards