0.1 Matrix inversion

Say that we have the following matrix and that we want to find its determinant.

Calculating the determinant of a matrix is a recursive process. Basically, we start by choosing any one row or column. The determinant will then befound with respect to this row or column. What this means is that we will find a sum of the products of this row or column's values andsub-determinants formed by blocking out the row and column of the particular value.

Why is this choice of row or column left to us instead of always being defined as, say, the first row? The reason is that by choosing this row orcolumn wisely, we can sometimes reduce the amount of work we do. For example, if a certain row or column contains a few zeros, choosing it as therow/column that we take the determinant with respect to would be a smart move. As the values of this chosen row or column will be multiplied bysub-determinants of the matrix in question, a value of $0$ in one of these products would mean that we have one less matrix whose determinant we needto calculate.

In the case of the matrix above, we'll compute the determinant with respect to the first column. The final equation for the determinant is:

Here, $A^{\mathrm{ij}}$ means the matrix formed by eliminating the $i$ -th column and the $j$ -th row of $A$ .

Let's just look at the first term in final equation for determinant . It is basically the first element of $A$ 's first column times the determinant of the matrix formed by the elimination of the first row and first column of $A$ . There is also a $-1^{(r+c)}$ term included. This serves to make the signs of all of the terms in the determinant equation fluctuate back and forth. The next term is the same, except that we have moved on to the second element in the first column of $A$ . As this element holds a position in the second row and first column of $A$ , the sub-determinant in this term is obtained by hiding the second row and first column of $A$ .

In a generic $3$ x $3$ example, we would find the following solution for the determinant:

To find the determinants of the $2$ x $2$ sub-determinants, we could again apply the rule of the final equation for determinant , keeping in mind that the determinant of a scalar value is simply that scalar value. However, it iseasier to remember the following solution