9.7 Solving systems with inverses (Page 5/8)

Precalculus

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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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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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