Name:________________________________
Date:________________________ Day:____
Algebra Chapter 4 Review – Sections 4.1, 4.2, 4.4, 4.5 only
Matrix Operations
−2 2
0 1
1
𝐴= [
], 𝐵 = [
], 𝐶 = [ ]
4 −1
−2 5
−2
Compute each of the following. If the operation is not possible, explain why.
1
3. 𝐵𝐴
1. 2 𝐴 − 𝐵
4. 𝐴𝐶
2. 𝐵 + 𝐶
Determinants
Calculate the determinant of each matrix.
8 −4
5. [
]
−6 3
2 3
6. [
]
−1 4
7. Explain a use for the determinant of a matrix.
Inverse Matrices
For numbers 8-10, calculate the inverse of each matrix. If the matrix does not have an inverse, explain why.
2
1
−4 −2
2
8. [
9. [
10. [ ]
]
]
−6 −3
5
2
−5
For numbers 11-12, use an inverse matrix to solve the system of equations.
3𝑥 + 2𝑦 = −8
𝑥−𝑦 =5
11. {
12. {
−2𝑥 + 5𝑦 = 18
−2𝑥 + 3𝑦 = −9
Real-World Applications
13. You have $18 to spend for lunch during a 5 day work week. It costs you $1.50 to make a lunch at home and
$5 to buy a lunch. How many times during the work week should you make a lunch at home in order to stay
within your budget?
a. Write a system of equations to represent the situation. Be sure to define your variables.
b. Solve the system of equations using Cramer´s rule or an inverse matrix. Show your work.
c. Answer the question.
14. On a test that contains multiple choice and free response questions, you answered a total of 12 questions
correctly. You earned 5 points for each correct multiple choice answer and 15 points for each correct freeresponse answer and earned a total of 80 points. How many multiple choice questions did you answer
correctly?
a. Write a system of equations to represent the situation. Be sure to define your variables.
b. Solve the system of equations using Cramer´s rule or an inverse matrix. Show your work.
c. Answer the question.
ANSWER KEY
−1
0
1. [
]
4 −5.5
2. Not possible because matrices do not have the same dimensions.
4 −1
3. [
]
24 −9
−6
4. [ ]
6
5. 0
6. 11
7. Determinants are used to find inverses of matrices and to solve systems of equations using Cramer’s rule.
8. 𝐷 = 9, 𝐷𝑥 = 45, 𝐷𝑦 = −9 so 𝑥 =
45
9
= 5, 𝑦 =
−9
9
= −1 Solution: (5, -1)
9. 𝐷 = 0, 𝐷𝑥 = −16, 𝐷𝑦 = −32 Since D = 0 but the other determinants are not zero, there is no solution.
10. No inverse because the determinant is 0
1
1
11. [
]
−2.5 −2
12. No inverse because only square matrices have inverses.
3 2 𝑥
−8
13. [
][ ] = [ ]
−2 5 𝑦
18
1 −1 𝑥
5
14. [
][ ] = [ ]
−2 3 𝑦
−9
𝐴−1 = [
5/19 −2/19
]
2/19 3/19
𝐴−1 = [
3 1
]
2 1
𝑥
−4
[𝑦] = 𝐴−1 𝐵 = [ ] Solution: (-4, 2)
2
𝑥
6
[𝑦] = 𝐴−1 𝐵 = [ ] Solution: (6, 1)
1
1.5𝑥 + 5𝑦 = 18
x = #times you make lunch, y = # times you buy lunch
𝑥+𝑦 =5
b. (2, 3)
c. You should make lunch twice and buy lunch 3 times.
15. a. {
5𝑥 + 15𝑦 = 80
x = # correct multiple choice questions, y = # correct free-response questions
𝑥 + 𝑦 = 12
b. (10, 2)
c. You answered 10 multiple choice questions correctly and 2 free-response questions correctly.
16. a. . {