# 9.7 Solving systems with inverses  (Page 5/8)

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Solve the system using the inverse of the coefficient matrix.

$X=\left[\begin{array}{c}4\\ 38\\ 58\end{array}\right]$

Given a system of equations, solve with matrix inverses using a calculator.

1. Save the coefficient matrix and the constant matrix as matrix variables $\text{\hspace{0.17em}}\left[A\right]\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left[B\right].$
2. Enter the multiplication into the calculator, calling up each matrix variable as needed.
3. If the coefficient matrix is invertible, the calculator will present the solution matrix; if the coefficient matrix is not invertible, the calculator will present an error message.

## Using a calculator to solve a system of equations with matrix inverses

Solve the system of equations with matrix inverses using a calculator

$\begin{array}{l}2x+3y+z=32\hfill \\ 3x+3y+z=-27\hfill \\ 2x+4y+z=-2\hfill \end{array}$

On the matrix page of the calculator, enter the coefficient matrix    as the matrix variable $\text{\hspace{0.17em}}\left[A\right],\text{\hspace{0.17em}}$ and enter the constant matrix as the matrix variable $\text{\hspace{0.17em}}\left[B\right].$

$\left[A\right]=\left[\begin{array}{ccc}2& 3& 1\\ 3& 3& 1\\ 2& 4& 1\end{array}\right],\text{ }\left[B\right]=\left[\begin{array}{c}32\\ -27\\ -2\end{array}\right]$

On the home screen of the calculator, type in the multiplication to solve for $\text{\hspace{0.17em}}X,\text{\hspace{0.17em}}$ calling up each matrix variable as needed.

${\left[A\right]}^{-1}×\left[B\right]$

Evaluate the expression.

$\left[\begin{array}{c}-59\\ -34\\ 252\end{array}\right]$

Access these online resources for additional instruction and practice with solving systems with inverses.

## Key equations

 Identity matrix for a $2\text{}×\text{}2$ matrix ${I}_{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ Identity matrix for a $\text{3}\text{}×\text{}3$ matrix ${I}_{3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ Multiplicative inverse of a $2\text{}×\text{}2$ matrix

## Key concepts

• An identity matrix has the property $\text{\hspace{0.17em}}AI=IA=A.\text{\hspace{0.17em}}$ See [link] .
• An invertible matrix has the property $\text{\hspace{0.17em}}A{A}^{-1}={A}^{-1}A=I.\text{\hspace{0.17em}}$ See [link] .
• Use matrix multiplication and the identity to find the inverse of a $\text{\hspace{0.17em}}2×2\text{\hspace{0.17em}}$ matrix. See [link] .
• The multiplicative inverse can be found using a formula. See [link] .
• Another method of finding the inverse is by augmenting with the identity. See [link] .
• We can augment a $\text{\hspace{0.17em}}3×3\text{\hspace{0.17em}}$ matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. See [link] .
• Write the system of equations as $\text{\hspace{0.17em}}AX=B,\text{\hspace{0.17em}}$ and multiply both sides by the inverse of $\text{\hspace{0.17em}}A:{A}^{-1}AX={A}^{-1}B.\text{\hspace{0.17em}}$ See [link] and [link] .
• We can also use a calculator to solve a system of equations with matrix inverses. See [link] .

## Verbal

In a previous section, we showed that matrix multiplication is not commutative, that is, $\text{\hspace{0.17em}}AB\ne BA\text{\hspace{0.17em}}$ in most cases. Can you explain why matrix multiplication is commutative for matrix inverses, that is, $\text{\hspace{0.17em}}{A}^{-1}A=A{A}^{-1}?$

If $\text{\hspace{0.17em}}{A}^{-1}\text{\hspace{0.17em}}$ is the inverse of $\text{\hspace{0.17em}}A,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}A{A}^{-1}=I,\text{\hspace{0.17em}}$ the identity matrix. Since $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is also the inverse of $\text{\hspace{0.17em}}{A}^{-1},{A}^{-1}A=I.\text{\hspace{0.17em}}$ You can also check by proving this for a $\text{\hspace{0.17em}}2×2\text{\hspace{0.17em}}$ matrix.

Does every $\text{\hspace{0.17em}}2×2\text{\hspace{0.17em}}$ matrix have an inverse? Explain why or why not. Explain what condition is necessary for an inverse to exist.

Can you explain whether a $\text{\hspace{0.17em}}2×2\text{\hspace{0.17em}}$ matrix with an entire row of zeros can have an inverse?

No, because $\text{\hspace{0.17em}}ad\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}bc\text{\hspace{0.17em}}$ are both 0, so $\text{\hspace{0.17em}}ad-bc=0,\text{\hspace{0.17em}}$ which requires us to divide by 0 in the formula.

Can a matrix with an entire column of zeros have an inverse? Explain why or why not.

Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a $\text{\hspace{0.17em}}2×2\text{\hspace{0.17em}}$ matrix.

Yes. Consider the matrix $\text{\hspace{0.17em}}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].\text{\hspace{0.17em}}$ The inverse is found with the following calculation: $\text{\hspace{0.17em}}{A}^{-1}=\frac{1}{0\left(0\right)-1\left(1\right)}\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].$

difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott