Revision 2
Question 1.
Let 𝐴 = [
2 −1
1 3
] 𝐵=[
] compute 𝐴 + 𝐵 and 𝐴 × 𝐵.
4 0
−2 5
Question 2.
Given 𝐴 = [
1 2
−1 3
3 1
0
] and 𝐵 = [2 4]
1
1 5
Determine if the addition is possible. If yes, compute 𝐴 + 𝐵.
Determine if the multiplication is possible. If yes, compute 𝐴 × 𝐵.
Question 3.
7
1
Find the transpose of the matrix 𝐴 = [3 −4]
0
5
Question 4.
Let 𝐴 and 𝐵 be two 2 × 2 matrices such that 𝐴𝐵 =0 (the zero matrix), but
neither 𝐴 nor 𝐵 is the zero matrix. Give an example of such matrices 𝐴 and 𝐵.
Question 5.
Determine if the matrix is invertible by calculating its determinant:
𝐴=[
1 2
]
3 −4
Question 6.
Given the matrix:
1 −3 2
𝐴 = [ 0 1 −5]
2 −7 4
a) Determine if the matrix is invertible using row reductions.
b) If invertible, give the inverse.
c) Check you answer by verifying 𝐴𝐴−1 = 𝐼.
Question 7.
4 7
1
Given a matrix 𝐴 = [
], solve the system 𝐴𝑥 = 𝑏 for 𝑏 = [ ] using the
2 6
0
inverse.
Question 8.
Decide if the following elementary matrices are invertible. If yes, find their
inverses:
a) 𝐸1 = swapping row 1 and row 2 of 𝐼2 .
b) 𝐸2 = multiplying row 2 of 𝐼2 by 3.
Question 9.
Write the matrix corresponding to the operation:
•
Add 2 times row 1 to row 3 of a 3 × 3 identity matrix. Is it an elementary
matrix?
Question 10.
Suppose 𝐴 is a 2 × 2 matrix with determinant 0.
•
Explain why 𝐴 cannot have an inverse, relating it to solving 𝐴𝑥 = 𝑏 for all
𝑏 ∈ ℝ2 .
Question 11.
Suppose 𝑇: ℝ2 → ℝ2 is a linear transformation with standard matrix
𝐴=[
1 4
]
2 7
a) Use the Invertible Matrix Theorem to decide if 𝑇 is invertible.
b) Find the matrix of the inverse transformation if it exists.
Question 12.
Let 𝐴 be a 3 × 3 matrix such that the system 𝐴𝑥 = 𝑏 has exactly one solution
for every 𝑏 ∈ ℝ3 .
•
Using the Invertible Matrix Theorem, list three properties that A must
satisfy.
Question 13.
2
0
Given the matrix 𝐴 = 0
0
0
[0
0
2
0
0
0
0
0
0
2
0
0
0
1
0
2
5
0
0
3
4
1
0
6
2
2
1
5
0
0
7]
Find the inverse using partition.
Question 14.
Given the partitioned matrix:
a) Find the block matrix form of 𝐴𝑇 .
b) Specify the dimension of each block after the transpose.
Question 15.
Find the LU factorization of the following 4 × 4 matrix:
2 3 1 5
4 7 5 8
𝐴=[
]
6 9 8 12
2 3 4 7
Question 16.
Suppose you have solved 𝐴𝑥 = 𝑏 where:
1 2
𝐴 = [2 5
4 10
1
6
2], 𝑏 = [15 ]
3
25
a) Find the LU factorization of 𝐴 (Without row exchanges).
b) Solve 𝐴𝑥 = 𝑏 using the previous factorization obtained.
Question 17.
Find the determinant of the matrix
2 0
𝐴 = [ −3 4
1 −1
1
2]
5
Question 18.
2
0
Given the following matrix 𝐴 = [
0
0
3
4
0
0
102
25
5
0
5
68
]
71
10
Calculate the 𝑑𝑒𝑡(𝐴).
Question 19.
2 0
𝐴 = [ −3 4
1 −1
1
1 2
2] and 𝐵 = [ 0 5
5
0 1
1
2]
3
Calculate det (𝐴𝐵), det(4𝐴).
Question 20.
Solve the system using Cramer’s Rule:
2𝑥1 − 3𝑥2 − 𝑥3 = 5
{ 𝑥1 − 2𝑥2 + 4𝑥3 = −2
3𝑥1 + 𝑥2 + 2𝑥3 = 7
a) Find the co-factors.
b) Give the Adjugate matrix of A.
c) Calculate the 𝑑𝑒𝑡𝐴.
d) Give the inverse.
Question 21.
Consider the point in ℝ3 (1,2,0), (2,5,1), (3,3,4) and (4,7,2).
Find the volume of the parallelepiped determined by these points.
Let 𝐸 be the solid region inside the cube:
0 ≤ 𝑥1 ≤ 1, 0 ≤ 𝑥2 ≤ 1 and 0 ≤ 𝑥3 ≤ 1
Apply the linear transformation:
2 0
𝑇(𝑥) = 𝐴𝑥 where 𝐴 = [ 0 3
0 0
Find the volume of the transformed region 𝑇(𝐸).
1
0]
4