Question

If \( A = \begin{pmatrix} 2 & x + 9 \\ 1 & 2x \end{pmatrix} \) is invertible, then \( x \neq \):

• A. 4
• B. 1
• C. 3
• D. 5

Hint:

To determine the values of \( x \) for which a matrix is invertible, compute the determinant of the matrix and solve for \( x \) such that the determinant is not zero.

Answer 1

The Correct Option is C

To determine the values of \( x \) for which the matrix \( A \) is invertible, we need to compute the determinant of \( A \). A matrix is invertible if and only if its determinant is non-zero.
The determinant of the matrix \( A = \begin{pmatrix} 2 & x + 9 \\ 1 & 2x \end{pmatrix} \) is given by: \[ \text{det}(A) = (2)(2x) - (1)(x + 9) \] Simplifying the expression: \[ \text{det}(A) = 4x - (x + 9) \] \[ \text{det}(A) = 4x - x - 9 = 3x - 9 \] For \( A \) to be invertible, the determinant must not be zero: \[ 3x - 9 \neq 0 \] Solving for \( x \): \[ 3x \neq 9 \] \[ x \neq 3 \] Therefore, the matrix \( A \) is invertible when \( x \neq 3 \).