Question

Let \[ A = \begin{pmatrix} 3 & -2 & 1 \\ -1 & 3 & -1 \end{pmatrix} \] and \[ B = \begin{pmatrix} 1 \\ \alpha \\ -1 \end{pmatrix}. \] If \[ AB = \begin{pmatrix} -2 \\ 6 \end{pmatrix}, \] then the value of \( \alpha \) is equal to:

• A. -1
• B. 1
• C. -2
• D. 2
• E. 0

Hint:

When solving for unknowns in matrix equations, always ensure to set up the matrix multiplication properly and equate corresponding elements to solve for the variables. This straightforward method helps in systematically determining each variable's value. Also, double-check your results by substituting the values back into the matrix equation to ensure the result matrix matches the given one.

Answer 1

The Correct Option is D

To find \( \alpha \), perform the matrix multiplication \( AB \).
Step 1: Compute the first element of \( AB \). \[ 3 \times 1 + (-2) \times \alpha + 1 \times (-1) = -2. \] Simplifying, we get: \[ 3 - 2\alpha - 1 = -2 \quad \Rightarrow \quad 2 - 2\alpha = -2 \quad \Rightarrow \quad -2\alpha = -4 \quad \Rightarrow \quad \alpha = 2. \] 
Step 2: Verify with the second element of \( AB \). \[ -1 \times 1 + 3 \times \alpha + (-1) \times (-1) = 6. \] Simplifying, we find: \[ -1 + 3 \times 2 + 1 = 6 \quad \Rightarrow \quad -1 + 6 + 1 = 6 \quad \Rightarrow \quad 6 = 6. \] The calculation confirms the correct value of \( \alpha \). 
Conclusion: The value of \( \alpha \) is \( 2 \), which matches option (D).