How can you find the inverse of a 2x2 matrix?

To find the inverse of a 2x2 matrix, one can use the formula that takes into account the determinant of the matrix. If you have a matrix represented as:

[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} ]

its inverse exists only when the determinant, calculated as ( det(A) = ad - bc ), is not equal to zero. If the determinant is non-zero, the inverse of matrix A can be computed using the formula:

[ A^{-1} = \frac{1}{det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} ]

This formula effectively reorganizes and negates certain elements of the original matrix and scales the whole by the reciprocal of its determinant. The role of the determinant is crucial; if it were zero, the inverse wouldn't exist since the matrix would be singular.

While other options suggest alternative methods or incorrect procedures, they do not provide a complete and accurate way to calculate the inverse of a 2x2 matrix as specified in the question. Therefore, utilizing the aforementioned formula when the determinant is non-zero is essential for determining the inverse of the matrix correctly.