COMPASSLearning

Talk to us Register Login

JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Matrices

Let

P = [\frac{3}{2} - \frac{1}{2} \frac{1}{2} \frac{3}{2}], A = [1011]

and

Q = PAP^{T}

. If

P^{T} Q^{2007} P = [a c b d]

, then

2 a + b - 3 c - 4 d

equal to

(A)2007
(B)2005✓ Correct
(C)2006
(D)2004

Answer: (B)

Solution

Matrix

P

is orthogonal (rotation matrix), so

P P^{T} = I

. Given

Q = P A P^{T}

, we have

P^{T} Q = P^{T} P A P^{T} = A P^{T}

, and

P^{T} QP = P^{T} P A P^{T} P = A

. Therefore

P^{T} Q^{2007} P = (P^{T} QP)^{2007} = A^{2007}

. For the nilpotent-like matrix

A = [1011]

, we have

A^{n} = [10 n 1]

. Thus

A^{2007} = [1020071]

, giving

a = 1, b = 2007, c = 0, d = 1

. Therefore

2 a + b - 3 c - 4 d = 2 + 2007 - 0 - 4 = 2005

.

JEE Main · 2023 · 08 Apr 2023 · 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 →