JEE Mathematics Question

JEE Question - Solution | JEE Mathematics

MCQMediumJEE 2023Determinants Basics

Correct answer:A

Standard Method

Given: The matrix $A$ is the $3 \times 3$ matrix shown in the figure.

Find: $|\operatorname{adj}(\operatorname{adj}(2A))|$ .

From the solution, the determinant of $A$ is evaluated by factoring factorial terms and reducing the determinant:

|A| = \frac{1}{5!6!7!} \begin{vmatrix} 1 & 6 & 42 \\ 1 & 7 & 56 \\ 1 & 8 & 72 \end{vmatrix}

Apply the row transformations shown:

R_3 \to R_3 - R_2, \qquad R_2 \to R_2 - R_1

which gives

\begin{vmatrix} 1 & 8 & 42 \\ 0 & 1 & 14 \\ 0 & 1 & 16 \end{vmatrix}

Hence, as stated in the solution,

|A| = 2

Now use the determinant property of adjugate for an $n \times n$ matrix:

|\operatorname{adj}(B)| = |B|^{n-1}

For $B = 2A$ and $n = 3$ ,

|2A| = 2^3 |A| = 2^3 \cdot 2 = 2^4

Therefore,

|\operatorname{adj}(2A)| = |2A|^{2} = (2^4)^2 = 2^8

Again applying the same property to $\operatorname{adj}(2A)$ ,

|\operatorname{adj}(\operatorname{adj}(2A))| = |\operatorname{adj}(2A)|^{2} = (2^8)^2 = 2^{16}

Therefore, the correct option is A.

Note: The solution heading says B, but the working clearly concludes $2^{16}$ , which matches option A.

Using $|\operatorname{adj}(B)| = |B|$ is incorrect. For an $n \times n$ matrix, the correct relation is $|\operatorname{adj}(B)| = |B|^{n-1}$ . Here $n = 3$ , so the exponent must be $2$ .
Forgetting that $|2A| \neq 2|A|$ for a $3 \times 3$ matrix is a common error. The correct rule is $|kA| = k^n |A|$ , so $|2A| = 2^3 |A|$ .
Stopping after finding $|\operatorname{adj}(2A)|$ gives only the intermediate result. The question asks for $|\operatorname{adj}(\operatorname{adj}(2A))|$ , so the adjugate determinant property must be applied one more time.

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