Discussion
If the matrix \(\begin{bmatrix} x & -3 \\ 2 & x - 5 \end{bmatrix}\) is singular, sum of the values of \(x\) is:
Answer: A
The sum of the values of \(x\) is 5.
A singular matrix has zero determinant:
\[ \begin{aligned} x(x - 5) - (-3)(2) &= 0 \\ x^{2} - 5x + 6 &= 0 \end{aligned} \]
The roots are \(x = 2\) and \(x = 3\), and their sum is 5 (equally, the sum of the roots of \(x^{2} - 5x + 6\) is the negative of the coefficient of \(x\)).
6 is the product of the roots, not their sum. 0 and \(-1\) cannot arise: whatever the individual roots, their sum is fixed at 5 by the coefficient of \(x\) in the quadratic.
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