The value of x for which the matrix \(\begin{bmatrix} 2 + x & 3 & 4 \\ 1 & -1 & 2 \\ x & 1 & -5 \end{bmatrix}\) is a singular matrix is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(-\displaystyle\frac{25}{13}\)
A square matrix is singular when its determinant is zero. Expanding along the first row:
\( \displaystyle \Delta = (2 + x)\begin{vmatrix} -1 & 2 \\ 1 & -5 \end{vmatrix} - 3\begin{vmatrix} 1 & 2 \\ x & -5 \end{vmatrix} + 4\begin{vmatrix} 1 & -1 \\ x & 1 \end{vmatrix} \)
\(\Delta = (2 + x)(5 - 2) - 3(-5 - 2x) + 4(1 + x)\)
\(\Delta = 6 + 3x + 15 + 6x + 4 + 4x\)
\(\Delta = 25 + 13x\)
Setting the determinant equal to zero:
\(25 + 13x = 0\)
\( \displaystyle x = -\frac{25}{13} \)