7.7 Solving systems with inverses  (Page 2/8)

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 $A$ is transformed into $I,$ the augmented matrix $I$ transforms into $A^{\mathrm{-1}}.$

For example, given

augment $A$ with the identity

Perform row operations    with the goal of turning $A$ into the identity.

1. Switch row 1 and row 2.
   $$\left[\begin{array}{rr}\hfill 5& \hfill 3\\ \hfill 2& \hfill 1\end{array}|\begin{array}{rr}\hfill 0& \hfill 1\\ \hfill 1& \hfill 0\end{array}\right]$$
2. Multiply row 2 by $\mathrm{-2}$ and add to row 1.
   $$\left[\begin{array}{rr}\hfill 1& \hfill 1\\ \hfill 2& \hfill 1\end{array}|\begin{array}{rr}\hfill \mathrm{-2}& \hfill 1\\ \hfill 1& \hfill 0\end{array}\right]$$
3. Multiply row 1 by $\mathrm{-2}$ and add to row 2.
   $$\left[\begin{array}{rr}\hfill 1& \hfill 1\\ \hfill 0& \hfill \mathrm{-1}\end{array}|\begin{array}{rr}\hfill \mathrm{-2}& \hfill 1\\ \hfill 5& \hfill \mathrm{-2}\end{array}\right]$$
4. Add row 2 to row 1.
   $$\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 0& \hfill \mathrm{-1}\end{array}|\begin{array}{rr}\hfill 3& \hfill \mathrm{-1}\\ \hfill 5& \hfill \mathrm{-2}\end{array}\right]$$
5. Multiply row 2 by $\mathrm{-1.}$
   $$\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 0& \hfill 1\end{array}|\begin{array}{rr}\hfill 3& \hfill \mathrm{-1}\\ \hfill \mathrm{-5}& \hfill 2\end{array}\right]$$

The matrix we have found is $A^{\mathrm{-1}}.$

Finding the multiplicative inverse of 2×2 matrices using a formula

When we need to find the multiplicative inverse of a $2\times 2$ matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.

If $A$ is a $2\times 2$ matrix, such as

the multiplicative inverse of $A$ is given by the formula

where $ad-bc\ne 0.$ If $ad-bc=0,$ then $A$ has no inverse.