Question

The matrix \( A = \begin{bmatrix} a & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix} \) is not invertible only if \( a = \)

• A. -17
• B. -16
• C. 16
• D. 17

Hint:

A matrix is not invertible if its determinant is zero. Always calculate the determinant to check if a matrix is invertible.

Answer 1

The Correct Option is A

Step 1: Calculate the determinant of the matrix.
The matrix is not invertible if its determinant is zero. We calculate the determinant of the matrix: \[ \text{det}(A) = a \begin{vmatrix} 0 & 1 \\ 1 & 2 \end{vmatrix} - (-1) \begin{vmatrix} -3 & 1 \\ -1 & 2 \end{vmatrix} + 4 \begin{vmatrix} -3 & 0 \\ -1 & 1 \end{vmatrix} \] After simplifying the determinant, we set it equal to zero and solve for \( a \), which gives \( a = -17 \).

Step 2: Conclusion.
Thus, the matrix is not invertible only if \( a = -17 \), corresponding to option (A).