Let αβ ≠ 0 and A = [{β, α, 3}, {α, α, β}, {-β, α, 2α}]. If B = [{3α, -9, 3α}, {-α, 7, -2α}, {-2α, 5, -2β}] is the matrix of cofactors

Question

Let \(\alpha \beta \neq 0\) and \(A=\left[\begin{array}{ccc}\beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2 \alpha\end{array}\right]\).

If \(B=\left[\begin{array}{ccc}3 \alpha & -9 & 3 \alpha \\ -\alpha & 7 & -2 \alpha \\ -2 \alpha & 5 & -2 \beta\end{array}\right]\) is the matrix of cofactors of the elements of \(A\), then \(\operatorname{det}(A B)\) is equal to :

(1) 343

(2) 125

(3) 64

(4) 216

Answer 1

Correct option is (4) 216

Equating co-factor of \(\mathrm{A}_{21}\)

\(\left(2 \alpha^{2}-3 \alpha\right)=\alpha\)

\(\alpha=0,2\) (accept)

Now, \(2 \alpha^{2}-\alpha \beta=3 \alpha\)

\(\alpha=2 , \beta=1\)

\(|\mathrm{AB}|=|\mathrm{A} \operatorname{cof}(\mathrm{A})|=|\mathrm{A}|^{3}\)

\(A=\left|\begin{array}{ccc}1 & 2 & 3 \\ 2 & 2 & 1 \\ -1 & 2 & 4\end{array}\right|=6-2(9)+3(6)=6\)

\(|A|^3 = 216\)