0.17 Matrix inversion

Let's say we have the square $n$ x $n$ matrix $A$ composed of real numbers. By "square", we mean it has the same numberof rows as columns.

The subscripts of the real numbers in this matrix denote the row and column numbers, respectively (i.e. $a_{1, 2}$ holds the position at the intersection of the first row and the second column).

We will denote the inverse of this matrix as $A^{(-1)}$ . A matrix inverse has the property that when it is multiplied by theoriginal matrix (on the left or on the right), the result will be the identity matrix.

To compute the inverse of $A$ , two steps are required. Both involve taking determinants ( $\det A$ ) of matrices. The first step is to find the adjoint ( $(A)$ ) of the matrix A. It is computed as follows:

where $A^{\mathrm{ij}}$ is the $(n-1)()$ x $(n-1)()$ matrix obtained from $A$ by eliminating its $i$ -th column and $j$ -th row . Note that we are not eliminating the $i$ -th row and $j$ -th column as you might expect.

To finish the process of determining the inverse, simply divide the adjoint by the determinant of the original matrix $A$ .