Academic Year 2021-2022
Linear Algebra for business (BCOR 111)
Midterm Exam
Time limit: 120 Minutes
Number of pages:8
Date: 31/03/2022
Student’s Family Name:
First Name:
C.I.N:
Group:
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Student’s Signature:
1. Candidates are allowed to use a calculator
2. You may not use books, notes or any electronic device including cell phones
3. Put answers in the blanks provided
4. You must show all of your work. An answer, right or wrong, without the proper
justification will receive little to no credit
Grade/100
Professors/Correctors
Name
1
Signature
MULTIPLE CHOICE QUESTIONS: [16 marks]
Part A State whether the following statements are Always (A) true, Sometimes (S) true, or
Never (N) true.
Circle one of A, S, N below. Justify your answer
1- If A and B are diagonal matrices then AB=BA
A
S
N
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2- If A and B are 𝑛 × 𝑛 matrices and A is singular , then (AB) is singular
A
S
N
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3- If 𝐴−1 = 𝐴𝑇 then 𝑑𝑒𝑡𝐴−1 = ±1
A
S
N
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Part B For each of the following statements, select the right alternative.
1) If matrix A is invertible, then which of the followings is NOT true? (a is any non-zero real
number scalar; 𝑛 is any positive integer)
a. (𝐴𝑇 )−1 = (𝐴−1 )𝑇
b. 𝐴 = 𝐴−1
c. (𝐴𝑛 )−1 = (𝐴−1 )𝑛
d. (𝑎𝐴)−1 =
1
𝑎
(𝐴−1 )
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2) If A and B are invertible 𝑛 × 𝑛 matrices then 𝐴(𝐵 −1 ) is also invertible
a. True
b. False
3) Let A and B be invertible 𝑛 × 𝑛 matrices
i-
(𝐴𝐵)𝑇 is also invertible
a-True
b-False
ii-
((𝐴𝐵)𝑇 )−1 = (𝐴−1 )𝑇 (𝐵−1 )𝑇
a-True
b-False
4) The rank of
a.
b.
c.
d.
1
[0
0
0
0
1
0
0
0
0
1
0
2
3] is
4
0
1
2
3
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Exercise 1 [16 Marks]
1-
1. Compute the following determinants
a- 𝑑𝑒𝑡(𝐴)
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b- 𝑑𝑒𝑡(2𝐴−1 )
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c- det(𝐴𝑇 𝐴)
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2. If A, B are 3 × 3 matrices with |𝐴| = 2 𝑎𝑛𝑑 |𝐵| = 3, compute |𝐴4 𝐵 𝑇 𝐴−1 |
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Exercise 2 [18 Marks]
Suppose that 𝑥1 , 𝑥2 , 𝑥3 satisfy
2𝑥1 − 3𝑥2 + 3𝑥3 = 1
2𝑥1 − 𝑥2 + 𝑥3 = 1
𝑥1 + 2𝑥2 + 𝑥3 = −2
Use Cramer’s rule to find the value of 𝑥2
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Exercise 3 [27 Marks]
A small manufacturing plant makes three types of inflatable boats: one-person, two-person, and
four-person models. Each boat requires the services of three departments, as listed in the table.
The cutting, assembly, and packaging departments have available a maximum of 380, 330, and
120 labor-hours per week, respectively.
Department
One-person Boat
Two-person Boat
Four-person Boat
Cutting
0.5 hr
1.0 hr
1.5 hr
Assembly
0.6 hr
0.9 hr
1.2 hr
Packaging
0.2 hr
0.3 hr
0.5 hr
How many boats of each type must be produced each week for the plant to operate at full
capacity? (Solve using Gaussian elimination and show all the steps)
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Exercise 4 [23 Marks]
Consider the following system of linear equations:
𝑥1 + 2𝑥3 + 4𝑥4 = −8
𝑥2 − 3𝑥3 − 𝑥4 = 6
{
3𝑥1 + 4𝑥2 − 6𝑥3 + 8𝑥4 = 0
−𝑥2 + 3𝑥3 + 4𝑥4 = −6
a- Find the reduced echelon form of the augmented matrix associated with this system
b- Write the general solution
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