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Matrix Inversion

Module by: Thanos Antoulas

Summary: How to invert matrices

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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

A=a11a12…a1na21a22…a2n⋮⋮⋱⋮an1an2…ann A a11 a12 … a1n a21 a22 … a2n ⋮ ⋮ ⋱ ⋮ an1 an2 … ann (1)

Calculating the determinant of a matrix is a recursive process. Basically, we start by choosing any one row or column. The determinant will then be found 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 and sub-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 or column 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 the row/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 by sub-determinants of the matrix in question, a value of 00 in one of these products would mean that we have one less matrix whose determinant we need to 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:

detA=a11-11+1detA11+a21-12+1detA12+…+an1-1n+1detA1n A a11 -1 1 1 A11 a21 -1 2 1 A12 … an1 -1 n 1 A1n (2)

Here, AijAij means the matrix formed by eliminating the ii-th column and the jj-th row of AA.

Let's just look at the first term in final equation for determinant. It is basically the first element of AA's first column times the determinant of the matrix formed by the elimination of the first row and first column of AA. There is also a -1r+c1rc 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 AA. As this element holds a position in the second row and first column of AA, the sub-determinant in this term is obtained by hiding the second row and first column of AA.

In a generic 33 x 33 example, we would find the following solution for the determinant:

deta11a12a13a21a22a23a31a32a33=a11deta22a23a32a33−a21deta12a13a32a33+a31deta12a13a22a23 a11 a12 a13 a21 a22 a23 a31 a32 a33 a11 a22 a23 a32 a33 a21 a12 a13 a32 a33 a31 a12 a13 a22 a23 (3)

To find the determinants of the 22 x 22 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 is easier to remember the following solution

detabcd=ad−bc a b c d ad bc (4)

Example 1

To clarify, take the following example of finding the determinant of a numeric 33 x 33 matrix.

A=1-12311-2-20 A 1 -1 2 3 1 1 -2 -2 0 (5)

First we need to choose a row or column to take the determinant with respect to. We notice that the element in the third row and third column is a zero. Knowing that choosing a row or column that contains a zero will reduce our workload, we will choose the third column. Then, by applying final equation for determinant, we get

detA=2×-14det31-2-2+1×-15det1-1-2-2+0 A 2 -1 4 3 1 -2 -2 1 -1 5 1 -1 -2 -2 0 (6)

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