Inverse Matrices

The inverse of a matrix is another matrix which, when multiplied by the first matrix, yields the unit matrix ${\bf I}$ (a matrix with all zeroes except 1's down the diagonal).

$${\bf A}^{-1} {\bf A} = {\bf I}.$$ (22)

In the general case, the inverse may be written

$${\bf A}^{-1} = \frac{1}{\vert{\bf A}\vert} {\bf A}_{cof}^T,$$ (23)

where ${\bf A}_{cof}^T$ is the transpose of the matrix of cofactors $A_{ij} = (-1)^{i+j} \vert{\bf M}_{ij}\vert$. For example:

$$\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)^... ...\left( \begin{array}{cc} d & -b \\ -c & a \end{array} \right).$$ (24)

Clearly, there are major problems in finding the inverse of a matrix if it has a determinant equal to zero, since the formula for ${\bf A}^{-1}$ involves dividing by the determinant. In such cases, we say that ${\bf A}$ is singular and has no inverse.