Let A and B be two square matrices of order 3 such that |A|=3 and |B|=2 . Then |A^ T A(adj(2 ~A))^ -1 (adj(4 ~B))(adj(AB))^ -1 AA^ T | is equal to :

Correct answer: D. 64 aligned & |A|=3,|~B|=2 & |~A^ T A(adj(2 ~A))^ -1 (adj(4 ~B))(adj(AB))^ -1 AA^ T | aligned ( array ccc =3 × 3 × (adj(2 ~A)^ -1 |×| adj(4 ~B)| ×|(adj(AB))^ -1 × 3 × 3. 1 |adj(2 ~A)| 2^ 12 × 2^2 1 |adj(AB)| array aligned & = 1 2^6|adj A| = 1 |adj B · adj A| & =(1)/(2^6 · 3^2) =(1)/(2^2 · 3^2) & =3^4 · (1)/(2^6 · 3^2) · 2^ 12 · 2^2 · (1)/(2^2 · 3^2)=64 & aligned

JEE Main 20245 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 5 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Matrices

Let A and B be two square matrices of order 3 such that |A|=3 and...

Let

A and B

be two square matrices of order 3 such that

|A|=3 and |B|=2

. Then \left|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}\right| is equal to :

Options

A108
B32
C81
D64

Correct answer

D. 64

Step-by-step solution

\begin{aligned} & |\mathrm{A}|=3,|\mathrm{~B}|=2 \\ & \left|\mathrm{~A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}\right|\end{aligned}

(

\begin{array}{ccc} =3 \times 3 \times \mid \left(\operatorname{adj}(2 \mathrm{~A})^{-1}|\times| \operatorname{adj}(4 \mathrm{~B})| \times|(\operatorname{adj}(\mathrm{AB}))^{-1}\mid \times 3 \times 3\right. \\ \downarrow \qquad\qquad \downarrow \qquad\qquad \downarrow \\ \frac{1}{|\operatorname{adj}(2 \mathrm{~A})|} \qquad 2^{12} \times 2^2 \qquad \frac{1}{|\operatorname{adj}(\mathrm{AB})|} \end{array}

\begin{aligned} & =\frac{1}{2^6|\operatorname{adj} \mathrm{A}|} \quad=\frac{1}{|\operatorname{adj} \mathrm{B} \cdot \operatorname{adj} \mathrm{A}|} \\ & =\frac{1}{2^6 \cdot 3^2} \quad=\frac{1}{2^2 \cdot 3^2} \\ & =3^4 \cdot \frac{1}{2^6 \cdot 3^2} \cdot 2^{12} \cdot 2^2 \cdot \frac{1}{2^2 \cdot 3^2}=64 \\ & \end{aligned}

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