5. Finding the Inverse of a Matrix

by M. Bourne

We need to find inverses of matrices so that we can solve systems of simultaneous equations.

(We'll see how to solve systems in the next section, Matrices and Linear Equations).

What are we doing?

If we multiply matrix A by the inverse of matrix A, we will get the identity matrix, I.

The concept of solving systems using matrices is similar to the concept of solving simple equations.

For example, to solve 7x = 14, we multiply both sides by the same number. We find the "inverse" of 7, which is 1/7. Multiplying both sides on the left by 1/7 gives:

(1/7) × 7x = (1/7) × 14

On the left, (1/7) × 7 = 1. The number 1 is the "identity" for multiplication of ordinary numbers. On the right, we get 2.

The solution for our equation is:

x = 2

We extend this concept of finding an inverse for solving a single equation, to solving systems of simultaneous equations.

We'll find the inverse of a matrix using 2 different methods. You can decide which one to use depending on the situation.

The first method is limited to finding the inverse of 2 × 2 matrices. It involves the use of the determinant of a matrix which we saw earlier.

Reminder: We can only find the determinant of a square matrix. For example, if A is the square matrix

then we can find the determinant of A:

= 10 + 3 = 13.

For convenience, we could have written the determinant of A as |A| and so our final answer would be:

|A| = 13

Method 1 - Transposing and Determinants

This method is only good for finding the inverse of a 2 × 2 matrix.

We'll see how this method works via an example.

Find the inverse, A^-1, of

using Method 1.

Answer

Let's see it in LiveMath:

LIVEMath

Method 2 - Adjunct Matrix (can be extended to any size)

NOTE: I have left Method 2 here for historical reasons. We will be using computers to find the inverse (or more importantly, the solution for the system of equations) of matrices larger than 2×2.

If you need to find the inverse of a 3×3 (or bigger) matrix using paper, then follow the steps given. It is tedious, but it will get you there. Good luck.

Method 2 uses the adjoint matrix method.

[Warning: This is long - and ancient history!]

Answer

Now let's see why it no longer makes sense to do processes like method 2 on paper...

LIVEMath

Inverses of Larger Matrices

Most real systems of equations are very large (up to 100 by 100 is common). We use computers to find these inverses. You need to understand what to give the computer and what it will give you as an answer.

Here is the LiveMath solution for the inverse of a 4 by 4 matrix.

LIVEMath

Why stop there? Let's look at a 5×5 inverse done using LiveMath:

LIVEMath

And we may as well do a 6×6 while we are at it:

LIVEMath

Exercise

Find the inverse of

by Method 1.

(I believe this is the level of inverse we should do on paper, so we get a sense of what an inverse is and how it may be calculated. Anything bigger than this should be done using computer :-)

Answer

There are other methods for finding inverses of matrices, including row operations. But oh, the tedium. Let's do it with computer.