If the matrix <span class="math-tex">\(\begin{bmatrix} -1 & 3 &\ \ \ 2 \\ \ \ \ 1 & \rm K & -3 \\ \ \ \ 1 & 4 & \ \ \ 5 \end{bmatrix}\)</span> has an inverse matrix, then the value of K is:
1. Given Condition for Invertibility: The determinant of the matrix must not be zero: \[ \det(A) \neq 0 \] 2. Calculate the Determinant: For the given matrix, the determinant is calculated as: \[ -1(5k + 12) - 3(5 - (-3)) + 2(4 - k) \neq 0 \] 3. Simplify the Expression: \[ -5k - 12 - 24 + 8 - 2k \neq 0 \] \[ -7k - 28 \neq 0 \] 4. Solve for \( k \): \[ -7k \neq 28 \] \[ k \neq -4 \] Therefore, the matrix is invertible for all values of \( k \) except: \[ \boxed{k \neq -4} \]
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