When working with the determinant of a scalar multiple of a matrix, the key idea is that multiplying a matrix \(A\) by a scalar \(k\) results in the determinant being multiplied by \(k^n\), where \(n\) is the size of the matrix. This property helps simplify the calculation of determinants when the matrix is scaled by a constant. For a \(4 \times 4\) matrix, raising the scalar to the power of 4 is crucial for finding the correct result.