Question

For which value of $ x $, the matrix $ A $ has no inverse where $$ A = \begin{pmatrix} 8 & x & 0 \\4 & 0 & 2 \\12 & 6 & 0 \end{pmatrix} $$

• A. 6
• B. 4
• C. 12
• D. 8

Hint:

A matrix has no inverse if and only if its determinant is zero. Always calculate the determinant to check for invertibility.

Answer 1

The Correct Option is B

The matrix \( A \) has no inverse if its determinant is zero. To find the determinant of \( A \), we calculate: \[ \text{det}(A) = \begin{vmatrix} 8 & x & 0 4 & 0 & 2 12 & 6 & 0 \end{vmatrix} \] Using cofactor expansion along the first row: \[ \text{det}(A) = 8 \begin{vmatrix} 0 & 2 6 & 0 \end{vmatrix} - x \begin{vmatrix} 4 & 2 \\ 12 & 0 \end{vmatrix} + 0 \] Now, calculate the 2x2 determinants: \[ \begin{vmatrix} 0 & 2 6 & 0 \end{vmatrix} = -12 \] \[ \begin{vmatrix} 4 & 2 12 & 0 \end{vmatrix} = -24 \]
Thus, we get: \[ \text{det}(A) = 8 \cdot (-12) - x \cdot (-24) = -96 + 24x \] For the matrix to have no inverse, we set the determinant equal to zero: \[ -96 + 24x = 0 \] Solving for \( x \): \[ 24x = 96 \] \[ x = 4 \]
Therefore, the correct answer is 2. 4.