11.7 Solving Systems with Inverses

Given matrix A, show that $A I = I A = A .$

Show that the given matrices are multiplicative inverses of each other.

Show that the following two matrices are inverses of each other.

Use matrix multiplication to find the inverse of the given matrix.

Use the formula to find the multiplicative inverse of

Use the formula to find the inverse of matrix $A .$ Verify your answer by augmenting with the identity matrix.

Find the inverse, if it exists, of the given matrix.

Given the $3 \times 3$ matrix $A ,$ find the inverse.

Find the inverse of the $3 \times 3$ matrix.

Solve the given system of equations using the inverse of a matrix.

Solve the following system using the inverse of a matrix.

Solve the system using the inverse of the coefficient matrix.

Solve the system of equations with matrix inverses using a calculator

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, $A^{−1} A = A A^{−1} ?$

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

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.

Show that matrix $A$ is the inverse of matrix $B .$

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} , B = \begin{bmatrix} −2 & 1 \\ \frac{3}{2} & - \frac{1}{2} \end{bmatrix}$

Show that matrix $A$ is the inverse of matrix $B .$

$A = \begin{bmatrix} −2 & \frac{1}{2} \\ 3 & −1 \end{bmatrix} , B = \begin{bmatrix} −2 & −1 \\ −6 & −4 \end{bmatrix}$

Show that matrix $A$ is the inverse of matrix $B .$

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 0 & 2 \\ 1 & 6 & 9 \end{bmatrix} , B = \frac{1}{4} \begin{bmatrix} 6 & 0 & −2 \\ 17 & −3 & −5 \\ −12 & 2 & 4 \end{bmatrix}$

Find the multiplicative inverse of each matrix, if it exists.

$\begin{bmatrix} 3 & −2 \\ 1 & 9 \end{bmatrix}$

Find the multiplicative inverse of each matrix, if it exists.

$\begin{bmatrix} −3 & 7 \\ 9 & 2 \end{bmatrix}$

Find the multiplicative inverse of each matrix, if it exists.

$\begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}$

Find the multiplicative inverse of each matrix, if it exists.

$\begin{bmatrix} 0.5 & 1.5 \\ 1 & −0.5 \end{bmatrix}$

Find the multiplicative inverse of each matrix, if it exists.

$\begin{bmatrix} 0 & 1 & −3 \\ 4 & 1 & 0 \\ 1 & 0 & 5 \end{bmatrix}$

Find the multiplicative inverse of each matrix, if it exists.

$\begin{bmatrix} 1 & 9 & −3 \\ 2 & 5 & 6 \\ 4 & −2 & 7 \end{bmatrix}$

Find the multiplicative inverse of each matrix, if it exists.

$\begin{bmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \end{bmatrix}$

Solve the system using the inverse of a $2 \times 2$ matrix.

$\begin{matrix}5 x - 6 y = - 61 \\ 4 x + 3 y = - 2\end{matrix}$

Solve the system using the inverse of a $2 \times 2$ matrix.

$\begin{matrix}3 x −2 y = 6 \\ - x + 5 y = −2\end{matrix}$

Solve the system using the inverse of a $2 \times 2$ matrix.

$\begin{matrix}−3 x −4 y = 9 \\ 12 x + 4 y = −6\end{matrix}$

Solve the system using the inverse of a $2 \times 2$ matrix.

$\begin{matrix}\frac{8}{5} x - \frac{4}{5} y = \frac{2}{5} \\ - \frac{8}{5} x + \frac{1}{5} y = \frac{7}{10}\end{matrix}$

Solve a system using the inverse of a $3 \times 3$ matrix.

$\begin{matrix}3 x −2 y + 5 z = 21 \\ 5 x + 4 y = 37 \\ x −2 y −5 z = 5\end{matrix}$

Solve a system using the inverse of a $3 \times 3$ matrix.

$\begin{matrix}6 x - 5 y - z = 31 \\ - x + 2 y + z = −6 \\ 3 x + 3 y + 2 z = 13\end{matrix}$

Solve a system using the inverse of a $3 \times 3$ matrix.

$\begin{matrix}4 x −2 y + 3 z = −12 \\ 2 x + 2 y −9 z = 33 \\ 6 y −4 z = 1\end{matrix}$

Solve a system using the inverse of a $3 \times 3$ matrix.

$\begin{matrix}\frac{1}{2} x - \frac{1}{5} y + \frac{1}{5} z = \frac{31}{100} \\ - \frac{3}{4} x - \frac{1}{4} y + \frac{1}{2} z = \frac{7}{40} \\ - \frac{4}{5} x - \frac{1}{2} y + \frac{3}{2} z = \frac{1}{4}\end{matrix}$

Use a calculator to solve the system of equations with matrix inverses.

$\begin{matrix}2 x - y = −3 \\ - x + 2 y = 2.3\end{matrix}$

Use a calculator to solve the system of equations with matrix inverses.

$\begin{matrix}12.3 x −2 y −2.5 z = 2 \\ 36.9 x + 7 y −7.5 z = −7 \\ 8 y −5 z = −10\end{matrix}$

Find the inverse of the given matrix.

$\begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{bmatrix}$

Find the inverse of the given matrix.

$\begin{bmatrix} 1 & - 2 & 3 & 0 \\ 0 & 1 & 0 & 2 \\ 1 & 4 & - 2 & 3 \\ - 5 & 0 & 1 & 1 \end{bmatrix}$

Find the inverse of the given matrix.

$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{bmatrix}$

Write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.

In the previous exercise, if you were told there were 400 more tickets sold for floor 2 than floor 1, how much was the price of each ticket?

Write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.

Students were asked to bring their favorite fruit to class. 95% of the fruits consisted of banana, apple, and oranges. If oranges were twice as popular as bananas, and apples were 5% less popular than bananas, what are the percentages of each individual fruit?

Write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.

A clothing store needs to order new inventory. It has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at $13.99, the beanie at $7.99, and the cowboy hat at $14.49. If 100 hats were sold this past quarter, $1,119 was taken in by sales,…

Write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.

Three roommates shared a package of 12 ice cream bars, but no one remembers who ate how many. If Micah ate twice as many ice cream bars as Joe, and Albert ate three less than Micah, how many ice cream bars did each roommate eat?

Write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.

Jay has lemon, orange, and pomegranate trees in his backyard. An orange weighs 8 oz, a lemon 5 oz, and a pomegranate 11 oz. Jay picked 142 pieces of fruit weighing a total of 70 lb, 10 oz. He picked 15.5 times more oranges than pomegranates. How many of each fruit did Jay pick?