Discussion
If
\[ \begin{pmatrix} -1 & 2 \\ 3 & -4 \\ -5 & 6 \end{pmatrix} \begin{pmatrix} 7 \\ 8 \end{pmatrix} = \begin{pmatrix} \alpha \\ \beta \\ 13 \end{pmatrix} \]
then the value of \(\alpha + \beta\) is equal to
Answer: E
\(\alpha + \beta = \) -2.
Multiply each row of the 3×2 matrix by the column (7, 8):
\[ \begin{aligned} \alpha &= (-1)(7) + 2(8) = 9 \\ \beta &= 3(7) + (-4)(8) = -11 \\ \text{row 3} &= (-5)(7) + 6(8) = 13 \end{aligned} \]
The third row reproduces the given 13, confirming the arithmetic. So \(\alpha + \beta = 9 - 11 = -2\). Sign slips in a single product produce values such as 18, 21 or -21.
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