- Find the inverse of a matrix when possible
- Solve a system of linear equations using an inverse matrix
In this section, we will explore how to solve systems of linear equations using the inverse of a matrix. This method is particularly useful for handling larger systems and provides a systematic and efficient approach to finding solutions. We will discuss the advantages of using matrix inversion and outline the steps involved in this process. Understanding this method will enhance your ability to solve complex systems and apply these techniques in various practical scenarios.
Finding the Inverse of a Matrix
We know that the multiplicative inverse of a real number [latex]a[/latex] is [latex]{a}^{-1}[/latex] and [latex]a{a}^{-1}={a}^{-1}a=\left(\frac{1}{a}\right)a=1[/latex]. For example, [latex]{2}^{-1}=\frac{1}{2}[/latex] and [latex]\left(\frac{1}{2}\right)2=1[/latex].
The multiplicative inverse of a matrix is similar in concept, except that the product of matrix [latex]A[/latex] and its inverse [latex]{A}^{-1}[/latex] equals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by [latex]{I}_{n}[/latex] where [latex]n[/latex] represents the dimension of the matrix. The equations below are the identity matrices for a [latex]2\text{}\times \text{}2[/latex] matrix and [latex]3\text{}\times \text{}3[/latex] matrix, respectively.
[latex]{I}_{2}=\left[\begin{array}{rrr}\hfill 1& \hfill & \hfill 0\\ \hfill 0& \hfill & \hfill 1\end{array}\right][/latex]
[latex]{I}_{3}=\left[\begin{array}{rrrrr}\hfill 1& \hfill & \hfill 0& \hfill & \hfill 0\\ \hfill 0& \hfill & \hfill 1& \hfill & \hfill 0\\ \hfill 0& \hfill & \hfill 0& \hfill & \hfill 1\end{array}\right][/latex]
The identity matrix acts as a [latex]1[/latex] in matrix algebra. For example, [latex]AI=IA=A[/latex].
A matrix that has a multiplicative inverse has the properties
[latex]\begin{array}{l}A{A}^{-1}=I\\ {A}^{-1}A=I\end{array}[/latex]
A matrix that has a multiplicative inverse is called an invertible matrix. Only a square matrix may have a multiplicative inverse, as the reversibility, [latex]A{A}^{-1}={A}^{-1}A=I[/latex], is a requirement. Not all square matrices have an inverse, but if [latex]A[/latex] is invertible, then [latex]{A}^{-1}[/latex] is unique. We will look at two methods for finding the inverse of a [latex]2\text{}\times \text{}2[/latex] matrix and a third method that can be used on both [latex]2\text{}\times \text{}2[/latex] and [latex]3\text{}\times \text{}3[/latex] matrices.
the identity matrix and multiplicative inverse
The identity matrix, [latex]{I}_{n}[/latex], is a square matrix containing ones down the main diagonal and zeros everywhere else.
[latex]\begin{array}{l}\begin{array}{l}\hfill \\ {I}_{2} & =\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 0& \hfill 1\end{array}\right]\begin{array}{cccc}& & & \end{array} & {I}_{3} & =\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\end{array}\right]\hfill \\ & \text{}2\times 2 & & \text{3}\times 3\hfill \end{array}\hfill \end{array}[/latex]
If [latex]A[/latex] is an [latex]n\times n[/latex] matrix and [latex]B[/latex] is an [latex]n\times n[/latex] matrix such that [latex]AB=BA={I}_{n}[/latex], then [latex]B={A}^{-1}[/latex], the multiplicative inverse of a matrix [latex]A[/latex].
- Given matrix [latex]A[/latex] of order [latex]n\times n[/latex] and matrix [latex]B[/latex] of order [latex]n\times n[/latex] multiply [latex]AB[/latex].
- If [latex]AB=I[/latex], then find the product [latex]BA[/latex]. If [latex]BA=I[/latex], then [latex]B={A}^{-1}[/latex] and [latex]A={B}^{-1}[/latex].
[latex]A=\left[\begin{array}{rrr}\hfill 1& \hfill & \hfill 5\\ \hfill -2& \hfill & \hfill -9\end{array}\right],B=\left[\begin{array}{rrr}\hfill -9& \hfill & \hfill -5\\ \hfill 2& \hfill & \hfill 1\end{array}\right][/latex]