GATE AG 2024 • Q36 MCQ • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Matrices and determinants

Question Statement:

$[P] = \begin{bmatrix} 2 & 3 & 7 \\ 4 & 1 & 5 \end{bmatrix}$ and $[Q] = \begin{bmatrix} 8 & 11 & 9 \\ 12 & 6 & 13 \end{bmatrix}$ are two matrices. The matrix $[P]^T [Q]$ is __________.

Options:

  • (A) $\begin{bmatrix} 64 & 36 & 116 \\ 46 & 39 & 107 \\ 70 & 40 & 128 \end{bmatrix}$
  • (B) $\begin{bmatrix} 64 & 46 & 70 \\ 36 & 39 & 40 \\ 116 & 107 & 128 \end{bmatrix}$
  • (C) $\begin{bmatrix} 112 & 113 \\ 88 & 89 \end{bmatrix}$
  • (D) $\begin{bmatrix} 112 & 88 \\ 113 & 89 \end{bmatrix}$

Official Answer Key:

B

Step-by-Step Solution & Explanation

Step 1 — check the order. P is 2 × 3, so Pᵀ is 3 × 2. Q is 2 × 3. The product (3 × 2)(2 × 3) is a 3 × 3 matrix — which immediately eliminates options (C) and (D). Step 2 — form Pᵀ (rows of Pᵀ are the columns of P): Pᵀ = [[2, 4], [3, 1], [7, 5]] Step 3 — multiply Pᵀ × Q: Row 1: (2×8 + 4×12, 2×11 + 4×6, 2×9 + 4×13) = (16+48, 22+24, 18+52) = (64, 46, 70) Row 2: (3×8 + 1×12, 3×11 + 1×6, 3×9 + 1×13) = (24+12, 33+6, 27+13) = (36, 39, 40) Row 3: (7×8 + 5×12, 7×11 + 5×6, 7×9 + 5×13) = (56+60, 77+30, 63+65) = (116, 107, 128) Result = [[64, 46, 70], [36, 39, 40], [116, 107, 128]] ⇒ (B) Trap: option (A) is the transpose of the correct answer.

Practice This in Real CBT Exam Interface

Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.

Launch CBT Mock Test