Stewart Precalc 6e Section 10.5: Inverses of Matrices and Matrix Equations

1 Concept - Identity and Inverse · Level 1

(a) The matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is called an _____ matrix. (b) If $A$ is a $2 \times 2$ matrix, then $A \times I = $ _____ and $I \times A = $ _____. (c) If $A$ and $B$ are $2 \times 2$ matrices with $A B = I$, then $B$ is the _____ of $A$.

2 Concept - Matrix Equation Solution · Level 1

(a) Write the system $5x + 3y = 4$, $3x + 2y = 3$ as a matrix equation $A X = B$. (b) Find $A^{-1}$. (c) Find $X = A^{-1} B$. (d) State the solution $x = $ _____, $y = $ _____.

3 Skills - Verifying Inverse Pair · Level 1

Calculate the products $A B$ and $B A$ to verify that $B$ is the inverse of $A$: $A = \begin{pmatrix} 4 & 1 \\ 7 & 2 \end{pmatrix}$, $B = \begin{pmatrix} 2 & -1 \\ -7 & 4 \end{pmatrix}$.

4 Skills - Verifying Inverse Pair · Level 1

Calculate the products $A B$ and $B A$ to verify that $B$ is the inverse of $A$: $A = \begin{pmatrix} 2 & -3 \\ 4 & -7 \end{pmatrix}$, $B = \begin{pmatrix} \dfrac{7}{2} & -\dfrac{3}{2} \\ 2 & -1 \end{pmatrix}$.

5 Skills - Verifying Inverse Pair (3x3) · Level 2

Calculate the products $A B$ and $B A$ to verify that $B$ is the inverse of $A$: $A = \begin{pmatrix} 1 & 3 & -1 \\ 1 & 4 & 0 \\ -1 & -3 & 2 \end{pmatrix}$, $B = \begin{pmatrix} 8 & -3 & 4 \\ -2 & 1 & -1 \\ 1 & 0 & 1 \end{pmatrix}$.

6 Skills - Verifying Inverse Pair (3x3) · Level 2

Calculate the products $A B$ and $B A$ to verify that $B$ is the inverse of $A$: $A = \begin{pmatrix} 3 & 2 & 4 \\ 1 & 1 & -6 \\ 2 & 1 & 12 \end{pmatrix}$, $B = \begin{pmatrix} 9 & -10 & -8 \\ -12 & 14 & 11 \\ -\dfrac{1}{2} & \dfrac{1}{2} & \dfrac{1}{2} \end{pmatrix}$.

7 Skills - Find Inverse and Verify · Level 2

Find the inverse of the matrix and verify that $A^{-1} A = A A^{-1} = I_2$: $A = \begin{pmatrix} 7 & 4 \\ 3 & 2 \end{pmatrix}$.

8 Skills - Find Inverse and Verify · Level 3

Find the inverse of the matrix and verify that $B^{-1} B = B B^{-1} = I_3$: $B = \begin{pmatrix} 1 & 3 & 2 \\ 0 & 2 & 2 \\ -2 & -1 & 0 \end{pmatrix}$.

9 Skills - Find Inverse (2x2) · Level 2

Find the inverse of the matrix if it exists: $\begin{pmatrix} -3 & -5 \\ 2 & 3 \end{pmatrix}$.

10 Skills - Find Inverse (2x2) · Level 2

Find the inverse of the matrix if it exists: $\begin{pmatrix} 3 & 4 \\ 7 & 9 \end{pmatrix}$.

11 Skills - Find Inverse (2x2) · Level 2

Find the inverse of the matrix if it exists: $\begin{pmatrix} 2 & 5 \\ -5 & -13 \end{pmatrix}$.

12 Skills - Find Inverse (2x2) · Level 2

Find the inverse of the matrix if it exists: $\begin{pmatrix} -7 & 4 \\ 8 & -5 \end{pmatrix}$.

13 Skills - Find Inverse (2x2) · Level 2

Find the inverse of the matrix if it exists: $\begin{pmatrix} 6 & -3 \\ -8 & 4 \end{pmatrix}$.

14 Skills - Find Inverse (2x2) · Level 2

Find the inverse of the matrix if it exists: $\begin{pmatrix} \dfrac{1}{2} & \dfrac{1}{3} \\ 5 & 4 \end{pmatrix}$.

15 Skills - Find Inverse (2x2) · Level 2

Find the inverse of the matrix if it exists: $\begin{pmatrix} 0.4 & -1.2 \\ 0.3 & 0.6 \end{pmatrix}$.

16 Skills - Find Inverse (3x3) · Level 3

Find the inverse of the matrix if it exists: $\begin{pmatrix} 2 & 4 & 1 \\ -1 & 1 & -1 \\ 1 & 4 & 0 \end{pmatrix}$.

17 Skills - Find Inverse (3x3) · Level 3

Find the inverse of the matrix if it exists: $\begin{pmatrix} 5 & 7 & 4 \\ 3 & -1 & 3 \\ 6 & 7 & 5 \end{pmatrix}$.

18 Skills - Find Inverse (3x3) · Level 3

Find the inverse of the matrix if it exists: $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & -1 \\ 1 & -1 & -10 \end{pmatrix}$.

19 Skills - Find Inverse (3x3) · Level 3

Find the inverse of the matrix if it exists: $\begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & 4 \\ 2 & 1 & 2 \end{pmatrix}$.

20 Skills - Find Inverse (3x3) · Level 3

Find the inverse of the matrix if it exists: $\begin{pmatrix} 0 & -2 & 2 \\ 3 & 1 & 3 \\ 1 & -2 & 3 \end{pmatrix}$.

21 Skills - Find Inverse (3x3) · Level 3

Find the inverse of the matrix if it exists: $\begin{pmatrix} 3 & -2 & 0 \\ 5 & 1 & 1 \\ 2 & -2 & 0 \end{pmatrix}$.

22 Skills - Find Inverse (4x4) · Level 4

Find the inverse of the matrix if it exists: $\begin{pmatrix} 1 & 2 & 0 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 2 & 0 & 2 \end{pmatrix}$.

23 Skills - Find Inverse (4x4) · Level 4

Find the inverse of the matrix if it exists: $\begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{pmatrix}$.

24 Skills - Solve System Using Inverse · Level 2

Solve the system $-3x - 5y = 4$, $2x + 3y = 0$ by converting to a matrix equation and using the inverse from Exercise 9.

25 Skills - Solve System Using Inverse · Level 2

Solve the system $3x + 4y = 10$, $7x + 9y = 20$ by using the inverse from Exercise 10.

26 Skills - Solve System Using Inverse · Level 2

Solve the system $2x + 5y = 2$, $-5x - 13y = 20$ by using the inverse from Exercise 11.

27 Skills - Solve System Using Inverse · Level 2

Solve the system $-7x + 4y = 0$, $8x - 5y = 100$ by using the inverse from Exercise 12.

28 Skills - Solve System Using Inverse · Level 3

Solve the system $2x + 4y + z = 7$, $-x + y - z = 0$, $x + 4y = -2$ by using the inverse from Exercise 17.

29 Skills - Solve System Using Inverse · Level 3

Solve the system $5x + 7y + 4z = 1$, $3x - y + 3z = 1$, $6x + 7y + 5z = 1$ by using the inverse from Exercise 18.

30 Skills - Solve System Using Inverse · Level 3

Solve the system $-2y + 2z = 12$, $3x + y + 3z = -2$, $x - 2y + 3z = 8$ by using the inverse from Exercise 21.

31 Skills - Solve 4-Variable System Using Inverse · Level 4

Solve the system $x + 2y + 3w = 0$, $y + z + w = 1$, $y + w = 2$, $x + 2y + 2w = 3$ by using the inverse from Exercise 23.

32 Skills - Calculator System Solution · Level 2

Use a calculator that can perform matrix operations to solve the system: $x + y - 2z = 3$, $2x + 5z = 11$, $2x + 3y = 12$.

33 Skills - Calculator System Solution · Level 2

Use a calculator that can perform matrix operations to solve the system: $3x + 4y - z = 2$, $2x - 3y + z = -5$, $5x - 2y + 2z = -3$.

34 Skills - Calculator System Solution · Level 3

Use a calculator to solve the system: $12x + \left(\dfrac{1}{2}\right)y - 7z = 21$, $11x - 2y + 3z = 43$, $13x + y - 4z = 29$.

35 Skills - Calculator System Solution · Level 3

Use a calculator to solve the system: $x + \left(\dfrac{1}{2}\right)y - \left(\dfrac{1}{3}\right)z = 4$, $x - \left(\dfrac{1}{4}\right)y + \left(\dfrac{1}{6}\right)z = 7$, $x + y - z = -6$.

36 Skills - Calculator 4-Variable System · Level 4

Use a calculator to solve the system: $x + y - 3w = 0$, $x - 2z = 8$, $2y - z + w = 5$, $2x + 3y - 2w = 13$.

37 Skills - Calculator 4-Variable System · Level 4

Use a calculator to solve the system: $x + y + z + w = 15$, $x - y + z - w = 5$, $x + 2y + 3z + 4w = 26$, $x - 2y + 3z - 4w = 2$.

38 Skills - Matrix Equation · Level 3

Solve the matrix equation $\begin{pmatrix} 3 & -2 \\ -4 & 3 \end{pmatrix} \begin{pmatrix} x & y & z \\ u & v & w \end{pmatrix} = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 1 & 3 \end{pmatrix}$ by multiplying each side by the appropriate inverse matrix.

39 Skills - Matrix Equation · Level 3

Solve the matrix equation $\begin{pmatrix} 0 & -2 & 2 \\ 3 & 1 & 3 \\ 1 & -2 & 3 \end{pmatrix} \begin{pmatrix} x & u \\ y & v \\ z & w \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 6 & 12 \\ 0 & 0 \end{pmatrix}$ by multiplying each side by the appropriate inverse matrix.

40 Skills - Inverse with Parameters · Level 3

Find the inverse of the matrix $\begin{pmatrix} a & -a \\ a & a \end{pmatrix}$ where $a \neq 0$.

41 Skills - Inverse of Diagonal Matrix · Level 3

Find the inverse of the diagonal matrix $\begin{pmatrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0 & d \end{pmatrix}$ where $a b c d \neq 0$.

42 Skills - Inverse with Parameter (Conditional Existence) · Level 3

Find the inverse of $\begin{pmatrix} 2 & x \\ x & x^2 \end{pmatrix}$. For what value(s) of $x$, if any, does the matrix have no inverse?

43 Skills - Inverse with Exponentials · Level 3

Find the inverse of $\begin{pmatrix} e^x & -e^{2x} \\ e^{2x} & e^{3x} \end{pmatrix}$. For what value(s) of $x$, if any, does the matrix have no inverse?

44 Skills - Inverse with Exponentials (3x3) · Level 4

Find the inverse of $\begin{pmatrix} 1 & e^x & 0 \\ e^x & -e^{2x} & 0 \\ 0 & 0 & 2 \end{pmatrix}$. For what value(s) of $x$, if any, does the matrix have no inverse?

45 Skills - Inverse with Parameter (Excluded Values) · Level 4

Find the inverse of $\begin{pmatrix} x & 1 \\ -x & 1/(x-1) \end{pmatrix}$. For what value(s) of $x$, if any, does the matrix have no inverse?

46 Application - Nutrition · Level 3

A nutritionist studies folic acid, choline, and inositol. Three food types contain per ounce: Type A: 3 mg folic acid, 4 mg choline, 3 mg inositol; Type B: 1 mg folic acid, 2 mg choline, 2 mg inositol; Type C: 3 mg folic acid, 4 mg choline, 4 mg inositol. (a) Find the inverse of $\begin{pmatrix} 3 & 1 & 3 \\ 4 & 2 & 4 \\ 3 & 2 & 4 \end{pmatrix}$. (b) How many ounces of each food are needed for 10 mg folic acid, 14 mg choline, and 13 mg inositol? (c) For 9 mg folic acid, 12 mg choline, and 10 mg inositol? (d) Will any combination supply 2 mg folic acid, 4 mg choline, and 11 mg inositol?

47 Application - Nutrition (Discussion) · Level 3

Refer to Exercise 47. Suppose Type C is mislabeled and actually contains 4 mg folic acid, 6 mg choline, and 5 mg inositol per ounce. Would it still be possible to use matrix inversion to solve parts (b), (c), and (d) of Exercise 47? Why or why not?

48 Application - Sales Commissions · Level 3

An encyclopedia saleswoman earns commission based on binding grade (standard, deluxe, leather). Week 1: 1 standard + 1 deluxe + 2 leather = \$675. Week 2: 2 standard + 1 deluxe + 1 leather = \$600. Week 3: 1 standard + 2 deluxe + 1 leather = \$625. (a) Set up a system in $x, y, z$ (commission per standard, deluxe, leather). (b) Express as a matrix equation $A X = B$. (c) Find $A^{-1}$ and solve.

49 Discovery - Zero-Product Property for Matrices · Level 4

Let $O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ be the $2 \times 2$ zero matrix. Find $2 \times 2$ matrices $A \neq O$ and $B \neq O$ such that $A B = O$. Can you find a matrix $A \neq O$ such that $A^2 = O$?

50 Example - Inverse of 2x2 Matrix · Level 2

Let $A = \begin{pmatrix} 4 & 5 \\ 2 & 3 \end{pmatrix}$. Find $A^{-1}$, and verify that $A A^{-1} = A^{-1} A = I_2$.

51 Example - Inverse of 3x3 Matrix · Level 3

Let $A = \begin{pmatrix} 1 & -2 & -4 \\ 2 & -3 & -6 \\ -3 & 6 & 15 \end{pmatrix}$. (a) Find $A^{-1}$. (b) Verify that $A A^{-1} = A^{-1} A = I_3$.

52 Example - Singular Matrix (No Inverse) · Level 3

Find the inverse of the matrix $\begin{pmatrix} 2 & -3 & -7 \\ 1 & 2 & 7 \\ 1 & 1 & 4 \end{pmatrix}$.

53 Example - Solving a System Using a Matrix Inverse · Level 3

A system of equations is given. (a) Write the system of equations as a matrix equation. (b) Solve the system by solving the matrix equation. $\begin{cases} 2x - 5y = 15 \\ 3x - 6y = 36 \end{cases}$

54 Example - Modeling Nutritional Requirements Using Matrix Equations · Level 3

A pet-store owner feeds his hamsters and gerbils different mixtures of three types of rodent food: KayDee Food, Pet Pellets, and Rodent Chow. Hamsters require 340 mg of protein, 280 mg of fat, and 440 mg of carbohydrates each day; gerbils need 480 mg of protein, 360 mg of fat, and 680 mg of carbohydrates each day. The amount of each nutrient (in mg) in one gram of each brand is: KayDee Food contains 10 mg protein, 10 mg fat, 5 mg carbohydrates; Pet Pellets contains 0 mg protein, 20 mg fat, 10 mg carbohydrates; Rodent Chow contains 20 mg protein, 10 mg fat, 30 mg carbohydrates. How many grams of each food should the storekeeper feed his hamsters and gerbils daily to satisfy their nutrient requirements exactly?

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