If \(A = \begin{bmatrix} 3 & 7 \\ 4 & -2 \end{bmatrix}, X = \begin{bmatrix} \alpha \\ -2 \end{bmatrix}, B = \begin{bmatrix} 7 \\ 32 \end{bmatrix}\) and \(AX = B\), then the value of the \(\alpha\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 7
By matrix multiplication, AX = B gives:
\( \displaystyle \begin{bmatrix} 3 & 7 \\ 4 & -2 \end{bmatrix}\begin{bmatrix} \alpha \\ -2 \end{bmatrix} = \begin{bmatrix} 3\alpha - 14 \\ 4\alpha + 4 \end{bmatrix} = \begin{bmatrix} 7 \\ 32 \end{bmatrix} \)
From the first row:
\(3\alpha - 14 = 7\)
\(3\alpha = 21,\quad \alpha = 7\)
The second row agrees: \(4(7) + 4 = 32\).