7.7 Solving systems with inverses (Page 5/8)

College algebra

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Try It

Solve the system using the inverse of the coefficient matrix.

\begin{array}{l} 2 x - 17 y + 11 z = 0 \\ - x + 11 y - 7 z = 8 \\ 3 y - 2 z = −2 \end{array}

$X = [\begin{matrix} 4 \\ 38 \\ 58 \end{matrix}]$

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How To

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

Save the coefficient matrix and the constant matrix as matrix variables $[A]$ and $[B] .$
Enter the multiplication into the calculator, calling up each matrix variable as needed.
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} 2 x + 3 y + z = 32 \\ 3 x + 3 y + z = −27 \\ 2 x + 4 y + z = −2 \end{array}

On the matrix page of the calculator, enter the coefficient matrix as the matrix variable $[A],$ and enter the constant matrix as the matrix variable $[B] .$

[A] = [\begin{matrix} 2 & 3 & 1 \\ 3 & 3 & 1 \\ 2 & 4 & 1 \end{matrix}], [B] = [\begin{matrix} 32 \\ −27 \\ −2 \end{matrix}]

On the home screen of the calculator, type in the multiplication to solve for $X,$ calling up each matrix variable as needed.

{[A]}^{−1} \times [B]

Evaluate the expression.

[\begin{matrix} −59 \\ −34 \\ 252 \end{matrix}]

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Media

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

Key equations

Identity matrix for a $2 \times 2$ matrix	$I_{2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
Identity matrix for a $3 \times 3$ matrix	$I_{3} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$
Multiplicative inverse of a $2 \times 2$ matrix	$A^{−1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}], where a d - b c \neq 0$

Key concepts

An identity matrix has the property $A I = I A = A .$ See [link] .
An invertible matrix has the property $A A^{−1} = A^{−1} A = I .$ See [link] .
Use matrix multiplication and the identity to find the inverse of a $2 \times 2$ 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 $3 \times 3$ 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 $A X = B,$ and multiply both sides by the inverse of $A : A^{−1} A X = A^{−1} B .$ See [link] and [link] .
We can also use a calculator to solve a system of equations with matrix inverses. See [link] .

Section exercises

Verbal

In a previous section, we showed that matrix multiplication is not commutative, that is, $A B \neq B A$ in most cases. Can you explain why matrix multiplication is commutative for matrix inverses, that is, $A^{−1} A = A A^{−1} ?$

If $A^{−1}$ is the inverse of $A,$ then $A A^{−1} = I,$ the identity matrix. Since $A$ is also the inverse of $A^{−1}, A^{−1} A = I .$ You can also check by proving this for a $2 \times 2$ matrix.

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Does every $2 \times 2$ matrix have an inverse? Explain why or why not. Explain what condition is necessary for an inverse to exist.

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Can you explain whether a $2 \times 2$ matrix with an entire row of zeros can have an inverse?

No, because $a d$ and $b c$ are both 0, so $a d - b c = 0,$ which requires us to divide by 0 in the formula.

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

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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 $2 \times 2$ matrix.

Yes. Consider the matrix $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] .$ The inverse is found with the following calculation: $A^{−1} = \frac{1}{0 (0) −1 (1)} [\begin{matrix} 0 & −1 \\ −1 & 0 \end{matrix}] = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] .$

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Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.

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While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma

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The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of

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Anatomy is the identification and description of the structures of living things

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Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.

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Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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