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

To find the inverse of a 2x2 matrix, you can use the formula based on the elements of the matrix. When given a matrix represented as:

[ \begin{pmatrix}

a & b \ c & d \end{pmatrix} ]

the formula for finding the inverse is derived from the condition that the product of a matrix and its inverse yields the identity matrix. The correct formula is:

[ \text{Inverse} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} ]

for which (ad - bc) must not be zero; if it is zero, the matrix does not have an inverse, as such a matrix is considered singular.

The reasoning behind this formula is that if you multiply the original matrix by this inverse, you will get the identity matrix, which confirms that the inverse is calculated correctly. The determinant (ad - bc) indicates whether the matrix is invertible; a non-zero determinant signifies that the matrix can indeed be inverted.

The answer provided captures the essential elements of this process, noting both the correct application of the formula and the importance of the determinant being