## How Do I Set Up and Solve a System with Inverses?

Writing a system of equations can be frustrating. In many cases, this starts when you do not write out which variables corresponds to what. How can you use “the smallest loan is one-half of the next larger loan” if you do not know which letter represents the amount of the  smaller loan and which letter represents the amount of the next larger loan?

Once you have the system, you can solve it with inverse matrices.

Problem 1 A bank gives three loans totaling 400,000 dollars to a development company for the purchase of three business properties. The largest loan is 100,000 dollars more than the sum of the other two, and the smallest loan is one-half of the next larger loan. Find the amount of each loan.

The key to writing out the equations for this problem is to make sure you know exactly which letter goes with which loan. Otherwise you don’t know whether to write x = 1/2y or y = 1/2x.

Once you have the solution (done with the inverse of A above), make sure it makes sense with the original problem statement. In the board below, the students solved the exact same problem using rref on their calculator. I expect that you will use some type of technology to do rref or find the inverse.

Problem 2 An investor has 400,000 dollars in three accounts, paying 6%, 8%, and 10%, respectively. If she has twice as much invested at 8% as she has at 6%, how much does she have invested in each account if she earns a total of 36,000 dollars in interest?

The second equation was originally y = 2x since the amount at 8% is twice the amount at 6%. This was then manipulated to put the system in a form where matrices can be used. Writing this equation out is MUCH simpler if you have written out what each variable represents somewhere (upper left) on the page.

## How Do You Find The Inverse Of A 2 x 2 Matrix?

Suppose the given square matrix is called A. To find the inverse of any matrix, we write the matrix in a larger matrix along side an identity matrix of the same size,

$latex \displaystyle \left[ \left. A\, \right|\,I \right]$

Now use row operations to rewrite this matrix so that the identity appears on the left side. The inverse of the original matrix will be on the right side of the transformed matrix,

$latex \displaystyle \left[ \left. I\, \right|\,{{A}^{-1}} \right]$

For instance, suppose we want to find the inverse of

$latex \displaystyle A=\left[ \begin{matrix} 2 & 2 \\ 2 & 1 \\ \end{matrix} \right]$

$latex \displaystyle \left[ \left. \begin{matrix} 2 & 2 \\ 2 & 1 \\ \end{matrix}\, \right|\,\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$

$latex \displaystyle \frac{1}{2}{{R}_{1}}\to {{R}_{1}}$

$latex \displaystyle \left[ \left. \begin{matrix} 1 & 1 \\ 2 & 1 \\ \end{matrix} \right|\begin{matrix} \frac{1}{2} & 0 \\ 0 & 1 \\ \end{matrix} \right]$

$latex \displaystyle -2{{R}_{1}}+{{R}_{2}}\to {{R}_{2}}$

$latex \displaystyle \left[ \left. \begin{matrix} 1 & 1 \\ 0 & -1 \\ \end{matrix} \right|\begin{matrix} \frac{1}{2} & 0 \\ -1 & 1 \\ \end{matrix} \right]$

$latex \displaystyle -1{{R}_{2}}\to {{R}_{2}}$

$latex \displaystyle \left[ \left. \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right|\begin{matrix} \frac{1}{2} & 0 \\ 1 & -1 \\ \end{matrix} \right]$

$latex \displaystyle -1{{R}_{2}}+{{R}_{1}}\to {{R}_{1}}$

$latex \displaystyle \left[ \left. \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right|\begin{matrix} -\frac{1}{2} & 1 \\ 1 & -1 \\ \end{matrix} \right]$

Let’s apply this strategy to finding a few more inverses.

Problem 1 Find the inverse of

$latex \displaystyle \left[ \begin{matrix} 2 & 4 \\ 2 & 5 \\ \end{matrix} \right]$

Problem 2 Find the inverse of

$latex \displaystyle \left[ \begin{matrix} 1 & 3 \\ 2 & 7 \\ \end{matrix} \right]$

## How Do I Find The Inverse Of A 2 x 2 Matrix?

Suppose the given square matrix is called A. To find the inverse of any matrix, we write the matrix in a larger matrix along side an identity matrix of the same size,

$latex \displaystyle \left[ \left. A\, \right|\,I \right]$

Now use row operations to rewrite this matrix so that the identity appears on the left side. The inverse of the original matrix will be on the right side of the transformed matrix,

$latex \displaystyle \left[ \left. I\, \right|\,{{A}^{-1}} \right]$