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]2text{}times text{}2[/latex] matrix and [latex]3text{}times text{}3[/latex] matrix, respectively.
[latex]{I}_{2}=left[begin{array}{rrr}hfill 1& hfill & hfill 0 hfill 0& hfill & hfill 1end{array}right][/latex]
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[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 1end{array}right][/latex]
The identity matrix acts as a 1 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=Iend{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]2text{}times text{}2[/latex] matrix and a third method that can be used on both [latex]2text{}times text{}2[/latex] and [latex]3text{}times text{}3[/latex] matrices.
Finding the Multiplicative Inverse Using Matrix Multiplication
We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication.
Finding the Multiplicative Inverse by Augmenting with the Identity
Another way to find the multiplicative inverse is by augmenting with the identity. When matrix [latex]A[/latex] is transformed into [latex]I[/latex], the augmented matrix [latex]I[/latex] transforms into [latex]{A}^{-1}[/latex].
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For example, given
[latex]A=left[begin{array}{rrr}hfill 2& hfill & hfill 1 hfill 5& hfill & hfill 3end{array}right][/latex]
augment [latex]A[/latex] with the identity
[latex]left[begin{array}{rr}hfill 2& hfill 1 hfill 5& hfill 3end{array}text{ }|text{ }begin{array}{rr}hfill 1& hfill 0 hfill 0& hfill 1end{array}right][/latex]
Perform row operations with the goal of turning [latex]A[/latex] into the identity.
- Switch row 1 and row 2. [latex]left[begin{array}{rr}hfill 5& hfill 3 hfill 2& hfill 1end{array}text{ }|text{ }begin{array}{rr}hfill 0& hfill 1 hfill 1& hfill 0end{array}right][/latex]
- Multiply row 2 by [latex]-2[/latex] and add to row 1. [latex]left[begin{array}{rr}hfill 1& hfill 1 hfill 2& hfill 1end{array}text{ }|text{ }begin{array}{rr}hfill -2& hfill 1 hfill 1& hfill 0end{array}right][/latex]
- Multiply row 1 by [latex]-2[/latex] and add to row 2. [latex]left[begin{array}{rr}hfill 1& hfill 1 hfill 0& hfill -1end{array}text{ }|text{ }begin{array}{rr}hfill -2& hfill 1 hfill 5& hfill -2end{array}right][/latex]
- Add row 2 to row 1. [latex]left[begin{array}{rr}hfill 1& hfill 0 hfill 0& hfill -1end{array}text{ }|text{ }begin{array}{rr}hfill 3& hfill -1 hfill 5& hfill -2end{array}right][/latex]
- Multiply row 2 by [latex]-1[/latex]. [latex]left[begin{array}{rr}hfill 1& hfill 0 hfill 0& hfill 1end{array}text{ }|text{ }begin{array}{rr}hfill 3& hfill -1 hfill -5& hfill 2end{array}right][/latex]
The matrix we have found is [latex]{A}^{-1}[/latex].
[latex]{A}^{-1}=left[begin{array}{rrr}hfill 3& hfill & hfill -1 hfill -5& hfill & hfill 2end{array}right][/latex]
Finding the Multiplicative Inverse of 2×2 Matrices Using a Formula
When we need to find the multiplicative inverse of a [latex]2times 2[/latex] matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.
If [latex]A[/latex] is a [latex]2times 2[/latex] matrix, such as
[latex]A=left[begin{array}{rrr}hfill a& hfill & hfill b hfill c& hfill & hfill dend{array}right][/latex]
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the multiplicative inverse of [latex]A[/latex] is given by the formula
[latex]{A}^{-1}=frac{1}{ad-bc}left[begin{array}{rrr}hfill d& hfill & hfill -b hfill -c& hfill & hfill aend{array}right][/latex]
where [latex]ad-bcne 0[/latex]. If [latex]ad-bc=0[/latex], then [latex]A[/latex] has no inverse.
Finding the Multiplicative Inverse of 3×3 Matrices
Unfortunately, we do not have a formula similar to the one for a [latex]2text{}times text{}2[/latex] matrix to find the inverse of a [latex]3text{}times text{}3[/latex] matrix. Instead, we will augment the original matrix with the identity matrix and use row operations to obtain the inverse.
Given a [latex]3text{}times text{}3[/latex] matrix
[latex]A=left[begin{array}{ccc}2& 3& 1 3& 3& 1 2& 4& 1end{array}right][/latex]
augment [latex]A[/latex] with the identity matrix
[latex]A|I=left[begin{array}{ccc}2& 3& 1 3& 3& 1 2& 4& 1end{array}text{ }|text{ }begin{array}{ccc}1& 0& 0 0& 1& 0 0& 0& 1end{array}right][/latex]
To begin, we write the augmented matrix with the identity on the right and [latex]A[/latex] on the left. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. We will find the inverse of this matrix in the next example.
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