Let A be a 3 × 3 matrix such that adj A = [ cases 2 & - 1 & 1 - 1 & 0 & 2 1 & - 2 & - 1 cases ] and B = adj (adjA) . If | A | = λ and | ( B ^ - 1 ) ^ ⊤ | = μ , then the ordered pair (|λ|, μ) is equal to

Correct answer: A. (3, (1)/(81)) |adjA| = |A| ^ 2 = 9 ⇒ |A| = ± 3 = λ ⇒ |λ| = 3 ⇒ |B| = |adjA| ^ 2 = 81 ⇒ | ( B ^ - 1 ) ^ T | = | B ^ - 1 | = |B| ^ - 1 = (1)/(|B|) = (1)/(81) = μ ∴ (|λ|, μ) = (3, (1)/(81)) .

JEE Main 20203 SeptemberEvening Shift

Exam: JEE Main
Year: 2020
Date: 3 September
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2020 · 3 September · Evening Shift · Mathematics · Matrices

Let A be a 3 × 3 matrix such that adj A = [ cases 2 & - 1 & 1 - 1 &...

Let A be a

3 \times 3

matrix such that

adj A = \left[

\begin{cases}2 & - 1 & 1 \\ - 1 & 0 & 2 \\ 1 & - 2 & - 1\end{cases}

\right] and

B = adj \left(adjA\right)

. If

| A | = \lambda and \left|{\left({B}^{- 1}\right)}^{⊤}\right| = \mu

, then the ordered pair \left(\left|\lambda\right|, \mu\right)

is equal to

Options

A $\left(3, \dfrac{1}{81}\right)$
B $\left(9, \dfrac{1}{9}\right)$
C $\left(3, 81\right)$
D $\left(9, \dfrac{1}{81}\right)$

Correct answer

A. $\left(3, \dfrac{1}{81}\right)$

Step-by-step solution

$\left|adjA\right| = {\left|A\right|}^{2} = 9$

$⇒ \left|A\right| = \pm 3 = \lambda ⇒ \left|\lambda\right| = 3$

$⇒ \left|B\right| = {\left|adjA\right|}^{2} = 81$

$⇒ \left|{\left({B}^{- 1}\right)}^{T}\right| = \left|{B}^{- 1}\right| = {\left|B\right|}^{- 1} = \dfrac{1}{\left|B\right|} = \dfrac{1}{81} = \mu$

$∴ \left(\left|\lambda\right|, \mu\right) = \left(3, \dfrac{1}{81}\right)$ .

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