Let α β neq 0 and A= [ begin matrix β & α & 3 \\α & α & β \\-β & α & 2α \\end matrix ] . If B= [ begin matrix 3α & -9 & 3α \\-α & 7 & -2α \\-2α & 5 & -2β \\end matrix ] is the matrix of cofactors of the elements of A , then det (A B) is equal to :

Mathematics · Matrices

JEE Main 2024 — 5 April, Shift 2 — Question 18

Let $\alpha \beta \neq 0$ and $A=\left[ \begin{matrix}\beta & \alpha & 3 \\\alpha & \alpha & \beta \\-\beta & \alpha & 2\alpha \\\end{matrix} \right]$ . If $B=\left[ \begin{matrix}3\alpha & -9 & 3\alpha \\-\alpha & 7 & -2\alpha \\-2\alpha & 5 & -2\beta \\\end{matrix} \right]$ is the matrix of cofactors of the elements of $A$ , then $\operatorname{det}(A B)$ is equal to :

Option A:
343
Option B:
125
Option C:
64
Option D:
216
Correct

Answer: D

Step-by-step solution

Equating co-factor fo $\mathrm{A}_{21}$

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

$\alpha=0,2$ (accept)

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

$\alpha=2 \quad \beta=1$

$|A B|=|A \operatorname{cof}(A)|=|A|^{3}$

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

$\left|A\right|=6$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 5 April, Shift 2
Subject: Mathematics
Chapter: Matrices
Topic: Adjoint of a Square Matrix