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Show that the following two matrices are inverses of each other.
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 .
Use matrix multiplication to find the inverse of the given matrix.
For this method, we multiply $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by a matrix containing unknown constants and set it equal to the identity.
Find the product of the two matrices on the left side of the equal sign.
Next, set up a system of equations with the entry in row 1, column 1 of the new matrix equal to the first entry of the identity, 1. Set the entry in row 2, column 1 of the new matrix equal to the corresponding entry of the identity, which is 0.
Using row operations, multiply and add as follows: $\text{\hspace{0.17em}}(\mathrm{-2}){R}_{1}+{R}_{2}\to {R}_{2}.\text{\hspace{0.17em}}$ Add the equations, and solve for $\text{\hspace{0.17em}}c.$
Back-substitute to solve for $\text{\hspace{0.17em}}a.$
Write another system of equations setting the entry in row 1, column 2 of the new matrix equal to the corresponding entry of the identity, 0. Set the entry in row 2, column 2 equal to the corresponding entry of the identity.
Using row operations, multiply and add as follows: $\text{\hspace{0.17em}}\left(\mathrm{-2}\right){R}_{1}+{R}_{2}={R}_{2}.\text{\hspace{0.17em}}$ Add the two equations and solve for $\text{\hspace{0.17em}}d.$
Once more, back-substitute and solve for $\text{\hspace{0.17em}}b.$
Another way to find the multiplicative inverse is by augmenting with the identity. When matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is transformed into $\text{\hspace{0.17em}}I,\text{\hspace{0.17em}}$ the augmented matrix $\text{\hspace{0.17em}}I\text{\hspace{0.17em}}$ transforms into $\text{\hspace{0.17em}}{A}^{\mathrm{-1}}.$
For example, given
augment $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ with the identity
Perform row operations with the goal of turning $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ into the identity.
The matrix we have found is $\text{\hspace{0.17em}}{A}^{\mathrm{-1}}.$
When we need to find the multiplicative inverse of a $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.
If $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is a $\text{\hspace{0.17em}}2\times 2\text{\hspace{0.17em}}$ matrix, such as
the multiplicative inverse of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is given by the formula
where $\text{\hspace{0.17em}}ad-bc\ne 0.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}ad-bc=0,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ has no inverse.
Use the formula to find the multiplicative inverse of
Using the formula, we have
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