Matrix inverses with Sage

Here are a few examples illustrating the computation of with Sage. After recalling the definition, we’ll show how to perform these computations directly and via the Reduced Row Echelon Form command

The definition

Recall that a square matrix \(M\) is called invertible if there’s a matrix \(M^{-1}\) of the same dimension such that \[MM^{-1} = I.\] In this case, \(M^{-1}\) is called the inverse of \(M\).

Example

Let \[ M = \left[\begin{array}{rrr} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 6 \end{array}\right]. \] Then \[ M^{-1} = \left[\begin{array}{rrr} -2 & 0 & 1 \\ 0 & 3 & -2 \\ 1 & -2 & 1 \end{array}\right], \] since \[ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 6 \end{array}\right] \left[\begin{array}{rrr} -2 & 0 & 1 \\ 0 & 3 & -2 \\ 1 & -2 & 1 \end{array}\right] = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \]

Computing inverses

You can find the inverse of a matrix \(M\) by setting up the augmented matrix \[[M|I],\] placing it in reduced row echelon form, and examining the results. If the \(M\) transforms to another copy of \(I\), then the \(I\) will have transformed to \(M^{-1}\). That is, \[ [M|I] \xrightarrow{RREF} [I|M^{-1}]. \] If \(M\) is not invertible, then you’ll find that \(M\) doesn’t row reduce to \(I\).

As a simple example to illustrate this second point, \[ \left[\begin{array}{rrrr} 1 & 2 & 1 & 0 \\ 2 & 4 & 0 & 1 \end{array}\right] \xrightarrow{RREF} \left[\begin{array}{rrrr} 1 & 2 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & -\frac{1}{2} \end{array}\right]. \] Thus, \[ \left[\begin{array}{rr} 1 & 2 \\ 2 & 4 \end{array}\right] \] is not invertible.

Using SageMath

Employing the strategy described above, we can compute of our first \(3\times3\) example using the following Sage code:

It’s worth mentioning that Sage matrices have a built in inverse method. Thus, we could compute, display, and check an inverse computation as follows: