Question

If the matrix \[ \begin{pmatrix} x & -3 \\ 2 & x - 5 \end{pmatrix} \] is singular, sum of the values of \( x \) is:

• A. \(5\)
• B. \(6\)
• C. \(0\)
• D. \(-1\)

Hint:

For singular matrices, always set the determinant equal to zero and solve for the roots. The sum of the roots can often give you the final answer, as in this case.

Answer 1

The Correct Option is A

For a matrix to be singular, its determinant must be zero. The determinant of a 2x2 matrix is given by: \[ {det}(A) = (x) \cdot (x - 5) - (-3) \cdot (2) \] Simplifying, we get: \[ {det}(A) = x(x - 5) + 6 \] \[ = x^2 - 5x + 6 \] Now, set the determinant equal to zero to make the matrix singular: \[ x^2 - 5x + 6 = 0 \] Factor the quadratic equation: \[ (x - 2)(x - 3) = 0 \] This gives the solutions: \[ x = 2 \quad {or} \quad x = 3 \] The sum of the values of \(x\) is: \[ 2 + 3 = 5 \] Thus, the sum of the values of \(x\) is \(5\), which corresponds to option (A).