If \(\begin{vmatrix} m-2 & m+1 \\ m+4 & m-2 \end{vmatrix} = -27\), find the value of m.

Answer Details

The determinant of a 2x2 matrix is given by the formula: \(\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc\) Using this formula, we can find the determinant of the given matrix: \(\begin{vmatrix} m-2 & m+1 \\ m+4 & m-2 \end{vmatrix} = (m-2)(m-2) - (m+1)(m+4)\) Simplifying this expression, we get: \((m-2)^2 - (m+1)(m+4) = m^2 - 4m + 4 - (m^2 + 5m + 4) = -9m\) Setting this equal to the given determinant of -27, we have: \(-9m = -27\) Solving for m, we get: \(m = 3\) Therefore, the answer is: \(3\).