Question

If \[ \begin{bmatrix} a & 2 & 3 b & 5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 2 3 & 4 -1 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 13 12 & 11 \end{bmatrix} \] then $(a,b)$ is

• A. $(1, -2)$
• B. $(-1, -4)$
• C. $(1, 3)$
• D. $(1, -4)$

Hint:

Always verify both columns — ensures no calculation mistake.

Answer 1

The Correct Option is D

Concept: Matrix multiplication: Row × Column.

Step 1: Multiply first row. \[ [a\ \ 2\ \ 3] \begin{bmatrix} 1 3 -1 \end{bmatrix} = a + 6 - 3 = a + 3 = 4 \Rightarrow a = 1 \] \[ [a\ \ 2\ \ 3] \begin{bmatrix} 2 4 1 \end{bmatrix} = 2a + 8 + 3 = 2a + 11 = 13 \Rightarrow a = 1 \text{ (consistent)} \]

Step 2: Multiply second row. \[ [b\ \ 5\ \ -1] \begin{bmatrix} 1 3 -1 \end{bmatrix} = b + 15 + 1 = b + 16 = 12 \Rightarrow b = -4 \] \[ [b\ \ 5\ \ -1] \begin{bmatrix} 2 4 1 \end{bmatrix} = 2b + 20 - 1 = 2b + 19 = 11 \Rightarrow b = -4 \text{ (consistent)} \]