Inverse Matrices

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 I_n.  Here are examples of the 4X4 and the 2X2 identity matrices.

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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 I_n, then MI_n = M.  Likewise I_mM = 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.

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.

The Process

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

1. Append the identity matrix I_n 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.