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

 Page 5 / 8

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: $\text{\hspace{0.17em}}{A}^{-1}=\frac{1}{0\left(0\right)-1\left(1\right)}\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].$

#### Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
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salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y