COMPASSLearning

Talk to us Register Login

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Matrices

Let A be a

3 \times 3

invertible matrix. If

∣ adj (24 A) ∣ = ∣ adj (3 adj (2 A)) ∣

, then

∣ A ∣^{2}

is equal to :

(A) $6^{6}$
(B) $2^{12}$
(C) $2^{6}$ ✓ Correct
(D)1

Answer: (C)

Solution

Use the property

∣ adj (M) ∣ = ∣ M ∣^{n - 1}

for an

n \times n

matrix.

For a

3 \times 3

matrix:

∣ adj (24 A) ∣ = (24)^{3} ∣ A ∣)^{2} = 2 4^{6} ∣ A ∣^{2}

.

Also,

∣ adj (3 adj (2 A)) ∣ = ∣3 adj (2 A) ∣^{2} = (3^{3} ∣ adj (2 A) ∣)^{2} = (3^{3} (2^{3} ∣ A ∣)^{2})^{2} = 3^{6} \cdot 2^{12} ∣ A ∣^{4}

.

Equating:

2 4^{6} ∣ A ∣^{2} = 3^{6} \cdot 2^{12} ∣ A ∣^{4}

.

Simplifying:

∣ A ∣^{2} = \frac{2 4 ^{6}}{3 ^{6} \cdot 2 ^{12}} = \frac{( 8 \cdot 3 ) ^{6}}{3 ^{6} \cdot 2 ^{12}} = \frac{2 ^{18}}{2 ^{12}} = 2^{6} = 64

.

Therefore,

∣ A ∣^{2} = 64 = 2^{6}

.

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Matrices →

More JEE Main questions on Matrices

For some $α, β \in R$ , let $A = [α 1 22]$ and $B = [11 1 β]$ be such that $A^{2} - 4 A + 2 I = B^{2} - 3 B + I = O$ . Then …
JEE Main 2026View solution →
For the matrices $A = [31 - 4 - 1]$ and $B = [- 29 - 13 4918]$ , if …
JEE Main 2026View solution →
If the system of equations
$3 x + y + 4 z = 3$
$2 x + α y - z = - 3$
$x + 2 y + z = 4$
has no solution, then the value of $α$ is equal to:
JEE Main 2026View solution →
If $A = [2335]$ , then the determinant of the matrix $(A^{2025} - 3 A^{2024} + A^{2023})$ is
JEE Main 2026View solution →
Let A be a $3 \times 3$ matrix such that $A + A^{T} = O$ . If …
JEE Main 2026View solution →
If $X = x y z$ is a solution of the system of equations $A X = B$ , where $adj A = 4 - 5 1 20 - 2 253$ and $B = 402$ , then $∣ x + y + z ∣$ is equal t …
JEE Main 2026View solution →