Let A be a matrix of order 3 × 3 and |A|=4 . If |2 adj(3 A)|=2^ a 3^ b , then what is the value of (α+β) ?

Correct answer: B. 13 |2 adj(3 A)|=2^3 ·|adj(3 A)| [ order of matrix =3]=2^3 ·|3 A|^ 3-1 =2^3 ·|3 A|^2=2^3(3^3)^2(|A|)^2=2^3 · 3^6 ·|A|^2=2^3 · 3^6 4^2=2^3 · 3^6 · 2^4=2^7 · 3^6 α=7, β=6α+β=13

NDA 2023

Exam: NDA
Year: 2023
Subject: Mathematics
Chapter: Matrices

NDA 2023 · Mathematics · Matrices

Let A be a matrix of order 3 × 3 and |A|=4 . If |2 adj(3 A)|=2^ a...

Let A be a matrix of order

3 \times 3

and

|\mathrm{A}|=4

. If

|2 \operatorname{adj}(3 \mathrm{ A})|=2^{\mathrm{a}} 3^{\mathrm{b}}

, then what is the value of

(\alpha+\beta)

Options

A $12$
B $13$
C $17$
D $24$

Correct answer

B. $13$

Step-by-step solution

|2 \operatorname{adj}(3 A)|=2^3 \cdot|\operatorname{adj}(3 A)| \quad[\because

order of matrix

=3]=2^3 \cdot|3 A|^{3-1}=2^3 \cdot|3 A|^2=2^3\left(3^3\right)^2(|A|)^2=2^3 \cdot 3^6 \cdot|A|^2=2^3 \cdot 3^6 4^2=2^3 \cdot 3^6 \cdot 2^4=2^7 \cdot 3^6\therefore \quad \alpha=7, \beta=6\alpha+\beta=13

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