Question

The value of \( x \) for which the inverse of the following matrix does not exist is \[ \begin{bmatrix} 1 & 3 & 0 \\ 2 & x & 4 \\ -1 & 0 & 2 \end{bmatrix} \]

• A. 0
• B. 1
• C. 10
• D. 12

Hint:

- A matrix is non-invertible (singular) if its determinant is zero.
- To find when a matrix is non-invertible, calculate its determinant and set it equal to zero.

Answer 1

The Correct Option is D

For a matrix to be non-invertible, its determinant must be zero. We calculate the determinant of the given matrix and solve for \( x \) when it equals zero. The determinant of a \(3 \times 3\) matrix is given by: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] For the given matrix: \[ A = \begin{bmatrix} 1 & 3 & 0 \\ 2 & x & 4 \\ -1 & 0 & 2 \end{bmatrix} \] Calculating the determinant: \[ \det(A) = 1 \cdot (x \cdot 2 - 4 \cdot 0) - 3 \cdot (2 \cdot 2 - 4 \cdot (-1)) + 0 \] \[ = 1 \cdot (2x) - 3 \cdot (4 + 4) \] \[ = 2x - 3 \cdot 8 \] \[ = 2x - 24 \] Now, for the matrix to be non-invertible, we set the determinant equal to zero: \[ 2x - 24 = 0 \] \[ 2x = 24 \quad \Rightarrow \quad x = 12 \] Thus, the value of \( x \) for which the inverse does not exist is: \[ \boxed{12} \]