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

Step 1: Understanding the Concept:
We need to perform matrix multiplication and then equate the corresponding elements of the resulting matrix with the given target matrix to solve for \( a \) and \( b \).
Step 2: Detailed Explanation:
Perform the multiplication: Let the result be matrix \( R \). The element \( R_{11} \) (first row, first column) is obtained by multiplying the first row of the first matrix with the first column of the second matrix: \[ R_{11} = a(1) + 2(3) + 3(-1) = a + 6 - 3 = a + 3 \] Equating to the given result \( R_{11} = 4 \): \[ a + 3 = 4 \implies a = 1 \] Now find \( b \) using element \( R_{21} \) (second row, first column): \[ R_{21} = b(1) + 5(3) + (-1)(-1) = b + 15 + 1 = b + 16 \] Equating to the given result \( R_{21} = 12 \): \[ b + 16 = 12 \implies b = -4 \] Thus, \( (a, b) = (1, -4) \).
Step 3: Final Answer:
The pair \( (a, b) \) is (1, -4).