5. Finding the Inverse of a Matrix

by M. Bourne

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 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).

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

`((2,3),(-1,5))`

then we can find the determinant of A:

`|(2,3),(-1,5)|=10+3=13`

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

`|A| = 13`

Another way of writing the same thing is to use "det" for "determinant". So for example, in this case we would write:

`det(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.

Example

Find the inverse, `A^-1`, of

`A=((2,-3),(4,-7))`

using Method 1.

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!]

Inverses of Larger Matrices (Method 3)

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.

However, some people need to know how to find inverses of large matrices!

See Inverse of a Matrix Using Gauss-Jordan Elimination for the most common method for finding inverses.

Exercise

Find the inverse of

`((7,-2),(-6,2))`

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 :-)

Let's now see some examples of products and inverses of matrices.

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