Problem-23

Question

Consider the following system of linear equations: \[ \begin{aligned} x_1 - 3x_2 &= 4\\ 3x_1 + kx_2 &= -12 \end{aligned} \]

where \(k \in \mathbb{R}\). If the given system has a unique solution, comment on the value that \(k\) cannot assume.

Solution

If \(A\) is a square matrix, then the system \(Ax = b\) has a unique solution when \(A\) is invertible. This happens when \(\text{det}(A) \neq 0\). Since we are asked to comment on the values that \(k\) cannot assume, we can set the determinant equal to zero: \[ \begin{vmatrix} 1 & -3\\ 3 & k \end{vmatrix} = 0 \implies k + 9 = 0 \implies k = -9 \] The system will have a unique solution as long as \(\boxed{k \neq -9}\).