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

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
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
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
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
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Abhi
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salma
Commplementary angles
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