Let |A|=6 , where A is a 3 × 3 matrix. If |adj(3 adj(~A^ 2 · adj(2 ~A)))|=2^ m · 3^ n , m, n ∈ N , then m+n is equal to \_\_\_\_ .

Correct answer: 0 For 3 × 3 matrix: |adj(M)| = |M|^2 , |kM| = k^3|M| . |2A| = 8 · 6 = 48 , |adj(2A)| = 48^2 = 2^8 · 3^2 . |A^2 · adj(2A)| = |A|^2 · |adj(2A)| = 2^2 · 3^2 · 2^8 · 3^2 = 2^ 10 · 3^4 . Let B = A^2 · adj(2A) . |adj(B)| = |B|^2 = 2^ 20 · 3^8 . |3\,adj(B)| = 3^3 · 2^ 20 · 3^8 = 2^ 20 · 3^ 11 . |adj(3\,adj(B))| = (2^ 20 · 3^ 11 )^2 = 2^ 40 · 3^ 22 . m = 40, n = 22 m + n = 62 .

JEE Main 202623 JanuaryMorning Shift

Exam: JEE Main
Year: 2026
Date: 23 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2026 · 23 January · Morning Shift · Mathematics · Matrices

Let |A|=6 , where A is a 3 × 3 matrix. If |adj(3 adj(~A^ 2 · adj(2...

Let

|\mathrm{A}|=6

, where A is a

3 \times 3

matrix. If \left|\operatorname{adj}\left(3 \operatorname{adj}\left(\mathrm{~A}^{2} \cdot \operatorname{adj}(2 \mathrm{~A})\right)\right)\right|=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}, \mathrm{m}, \mathrm{n} \in \mathbf{N}

, then

\mathrm{m}+\mathrm{n}

is equal to

\_\_\_\_$.

Correct answer

Step-by-step solution

For

3 \times 3

matrix:

|\text{adj}(M)| = |M|^2

|kM| = k^3|M|

|2A| = 8 \cdot 6 = 48

|\text{adj}(2A)| = 48^2 = 2^8 \cdot 3^2

|A^2 \cdot \text{adj}(2A)| = |A|^2 \cdot |\text{adj}(2A)| = 2^2 \cdot 3^2 \cdot 2^8 \cdot 3^2 = 2^{10} \cdot 3^4

.
Let

B = A^2 \cdot \text{adj}(2A)

|\text{adj}(B)| = |B|^2 = 2^{20} \cdot 3^8

|3\,\text{adj}(B)| = 3^3 \cdot 2^{20} \cdot 3^8 = 2^{20} \cdot 3^{11}

|\text{adj}(3\,\text{adj}(B))| = (2^{20} \cdot 3^{11})^2 = 2^{40} \cdot 3^{22}

m = 40, n = 22 \Rightarrow m + n = 62

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