Question

The matrix \[ \begin{bmatrix} 3 - x & 2 & 2 \\ 2 & 4 - x & 1 \\ -2 & -4 & -1 - x \end{bmatrix} \] is singular for the following values of \(x\).

• A. \( x = 0 \) and \( x = 3 \)
• B. \( x = 0 \) and \( x = -3 \)
• C. \( x = 0 \) and \( x = 6 \)
• D. \( x = 0 \) and \( x = -6 \)

Hint:

To determine when a matrix is singular, compute the determinant and set it equal to zero. The values of \(x\) that satisfy this condition make the matrix singular.

Answer 1

The Correct Option is A

A matrix is singular if its determinant is zero. The determinant of a 3x3 matrix \[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \] is given by: \[ \text{det} = a(ei - fh) - b(di - fg) + c(dh - eg) \] For the given matrix, we calculate the determinant: \[ \text{det} = (3 - x) \cdot \text{det}\left( \begin{bmatrix} 4 - x & 1 \\ -4 & -1 - x \end{bmatrix} \right) - 2 \cdot \text{det}\left( \begin{bmatrix} 2 & 1 \\ -2 & -1 - x \end{bmatrix} \right) + 2 \cdot \text{det}\left( \begin{bmatrix} 2 & 4 - x \\ -2 & -4 \end{bmatrix} \right) \] After performing the necessary calculations, you will find that the determinant is zero when \( x = 0 \) and \( x = 3 \), which means the matrix is singular for these values of \(x\). Thus, the correct answer is (A) \( x = 0 \) and \( x = 3 \), and (C) \( x = 0 \) and \( x = 6 \) as the matrix is singular for these values of \(x\).