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Inverse Matrices

Author: c o
Description:

To introduce the concept of inverse matrices
To demonstrate a method by which inverses of square matrices may be determined
To practice that method by working through an example

The identity matrix is first introduced and used to define the notion of invertible and singular matrices. A shortcut to finding the inverses of 2x2 matrices is then given. The relevance of invertible matrices is discussed with relation to matrix equations and linear systems. Next it is shown how Gaussian elimination may be used to determine the inverse of any invertible square matrix, and an example of doing so is provided in a slideshow.

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Tutorial

Terms and Notions

Background Material

Before beginning with this lesson, it might be a good idea to review packets on Matrix Multiplication, and Gaussian Elimination.  You might also view these examples of Gaussian Elimination.

The Identity Matrix

An nXn square matrix that that contains only 1's along its diagonal is called an identity matrix, and is denoted by In.  Here are examples of the 4X4 and the 2X2 identity matrices.

,

Identity matrices are so-called because they act as the multiplicative identity for matrix multiplication.  That is, given a matrix an mXn matrix M and given the identity matrix In, then MIn = M.  Likewise ImM = M.

Invertible and Singular Matrices

Any matrix that is not invertible is said to be singular.  Invertible matrices are interesting because any matrix equation involving one has a unique solution.  We will see more about this in the next section.

Some Properties Of Invertible Matrices

A Shortcut for 2x2 Matrices

When working with 2X2 matrices, there is a convenient shortcut for determining an inverse.

An important consequence of this short cut is that it gives us a good way to determine whether or not a matrix is invertible.  If the term ad - bc = 0, then there will be a zero on the denominator of 1/(ad-bc) in which case the inverse is undefined.

Connection With Matrix Equations

So any matrix equation, and also any square linear system, may be solved if we know the inverse of the matrix by which we are multiplying.  Of course, this only works with invertible matrices.

Finding Inverse Matrices

The Process

In order to find the inverse of an nXn matrix A, we take the following steps:

1. Append the identity matrix In onto the right of the of A producing a nX2n matrix (n rows and 2n columns)
2. Row reduce this new matrix using Gauss-Jordan Elimination
3. Take the rightmost n columns and use these to form a new matrix, this will be A-1.

For example if we start with this 2X2 matrix, and append I2 like this:

Then by applying Gauss-Jordan elimination, we'll end up with a 2X4 matrix that has I2 on the left side, and some other matrix on the right.  We take the right hand side and end up with the inverse of our starting matrix.

And now an example.

An Example

In this example we find the inverse of a 3x3 matrix using Gaussian elimination.