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

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
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ok
Zander
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Benetta
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lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
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Chris
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Momo
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the range is twice of the natural number which is the domain
Morolake
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6000
Robert
more than 6000
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can I see the picture
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with doing calculus
SLIMANE
Thanks po.
Jenica
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Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
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SLIMANE
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Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations