Applied Discrete Structures

In Chapter 5 we defined the inverse of an $n\times n$ matrix. We noted that not all matrices have inverses, but when the inverse of a matrix exists, it is unique. This enables us to define the inverse of an $n\times n$ matrix $A$ as the unique matrix $B$ such that $AB=BA=I\text{,}$ where $I$ is the $n\times n$ identity matrix. In order to get some practical experience, we developed a formula that allowed us to determine the inverse of invertible $2\times 2$ matrices. We will now use the Gauss-Jordan procedure for solving systems of linear equations to compute the inverses, when they exist, of $n\times n$ matrices, $n\ge 2\text{.}$ The following procedure for a $3\times 3$ matrix can be generalized for $n\times n$ matrices, $n\ge 2\text{.}$

Given the matrix $A=\left(\begin{array}{ccc}1& 1& 2\\ 2& 1& 4\\ 3& 5& 1\end{array}\right)\text{,}$ we want to find its inverse, the matrix $B=\left(\begin{array}{ccc}{x}_{11}& x_{12}& x_{13}\\ x_{21}& x_{22}& x_{23}\\ x_{31}& x_{32}& x_{33}\end{array}\right)\text{,}$ if it exists, such that $AB=I$ and $BA=I\text{.}$ We will concentrate on finding a matrix that satisfies the first equation and then verify that B also satisfies the second equation.

As the following theorem indicates, the verification that $BA=I$ is not necessary. The proof of the theorem is beyond the scope of this text. The interested reader can find it in most linear algebra texts.

It is clear from Chapter 5 and our discussions in this chapter that not all $n\times n$ matrices have inverses. How do we determine whether a matrix has an inverse using this method? The answer is quite simple: the technique we developed to compute inverses is a matrix approach to solving several systems of equations simultaneously.