Further practice: Linear Algebra and its applications - Gilbert Strang Linear Algebra and its applications - David C. Lay, Steven R. Lay, Judi J. McDonald

1. 1
Miguel Fernandes
LINEAR ALGEBRA
Exercises in
Matrices and Determinants
1. Consider the following matrix
𝑀 = (
2 1
21 3
1 91
)
1.1. Write:
1.1.1. The size of 𝑀.
1.1.2. The second column of 𝑀.
1.1.3. The first row of 𝑀.
1.1.4. The (2,2) entry of 𝑀.
1.2. Say whether the matrix 𝑀 is square. Justify your answer.
2. Write a 3 × 3 matrix such that:
 The entries are all different;
 The sum of the diagonal elements (usually called trace) is equal to 20;
 For 𝑖, 𝑗 ∈ {1,2,3}, the number
𝑥
𝑖+𝑗
is an integer, where 𝑥 is the entry (𝑖, 𝑗).
3. Find a 2 × 2 matrix 𝑋 such that 𝑋2
is diagonal but not 𝑋.
4. Let
𝐴 = (
12 3 1
2 0 2
18 0 1
), 𝐵 = (
5 5
9 9
1 4
) and 𝐶 = (
1 10 0
0 21 1
).
4.1. Find, if possible:
a) 𝐴 + 𝐵
b) 3𝐵
c) 3𝐵 + 𝐴
d) 𝐵 + 𝐶
e) 𝐴𝐵
f) 𝐴𝐼3
g) 𝐵𝐶
h) 𝐶𝐵
i) 3(𝐴𝐵)𝐶
4.2. Find a matrix 𝐷 with no zero entries such that 𝐴𝐷 is the 3 × 3 zero matrix.
5. Let 𝐴 be a square matrix that satisfies 𝐴2
+ 3𝐴 − 𝐼 = 0. Show that 𝐴3
= 10𝐴 − 3𝐼.
6. Consider two matrices 𝐴 and 𝐵. Show that if 𝐴 has a zero row, then 𝐴𝐵 has a zero row.
7. Recall the definition of trace of a matrix (exercise 2.) which we denote by 𝑡𝑟( ∙ ).
7.1. Find the trace of the matrix below.
𝐴 = (
3 3 2
0 1 2
1 2 1
)
7.2. Given two square matrices 𝐴 and 𝐵, show that:
7.2.1. 𝑡𝑟(𝐴 + 𝐵) = 𝑡𝑟(𝐴) + 𝑡𝑟(𝐵).
7.2.2. 𝑡𝑟(𝑘𝐴) = 𝑘𝑡𝑟(𝐴), 𝑘 ∈ ℝ.
7.2.3. 𝑡𝑟(𝐴) = 𝑡𝑟(𝐴 𝑇).
7.2.4. 𝑡𝑟(𝐴𝐵) = 𝑡𝑟(𝐵𝐴).
8. Let 𝐴 = (
0 0 0
𝑎 0 0
0 𝑏 0
) for some real numbers 𝑎 and 𝑏. Show that 𝐴 𝑛
= 0, for 𝑛 ≥ 3.
9. Let 𝐴 = (
8 3
2 8
). Find all the matrices that commute with 𝐴.

2. 2
Miguel Fernandes
10. Prove that the product of two diagonal matrices is diagonal.
11. Prove that the product of two upper triangular matrices is upper triangular.
12. Let 𝐴 = (
𝑎 ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 𝑏
) be a matrix (only the diagonal elements can be different from 0).
Show that 𝐴 𝑛
= (
𝑎 𝑛
⋯ 0
⋮ ⋱ ⋮
0 ⋯ 𝑏 𝑛
).
13. Consider the rotation matrix:
𝑅(𝛼) = (
cos 𝛼 − sin 𝛼
sin 𝛼 cos 𝛼
), for 𝛼 ∈ ℝ.
Verify that 𝑅(𝛼)𝑅(𝛽) = 𝑅(𝛼 + 𝛽), for 𝛼, 𝛽 ∈ ℝ. In particular, make 𝛽 = −𝛼 and say what matrix is 𝑅(𝛼 + 𝛽).
Think about 𝑥𝑦 plane and discuss what these results mean geometrically.
14. Find the transpose of each matrix of the exercise 4.
15. Given a square matrix 𝐴, show that 𝐴 + 𝐴 𝑇
, 𝐴𝐴 𝑇
and 𝐴 𝑇
𝐴 are symmetric.
16. Given a square matrix 𝐴, show that 𝐴 𝑇
− 𝐴 is skew-symmetric.
17. Is the following matrix Hermitian? Justify your answer.
𝐴 = (
1 3 − 2𝑖 1 − 𝑖
3 + 2𝑖 5 0
1 + 𝑖 0 7
)
18. Let 𝐴 and 𝐵 be square matrices. Show that if 𝐴𝐵 = 𝐴 and 𝐵𝐴 = 𝐵, then 𝐴2
= 𝐴, that is, 𝐴 is idempotent.
19. Show that, in a symmetric matrix of order 𝑛, we have
𝑛+𝑛2
2
entries that can be chosen independently.
What if we consider a skew-symmetric matrix?
20. Reduce the following matrices to row echelon form and find their ranks.
a) 𝐴 = (
1 4 7
2 5 8
3 6 9
)
b) 𝐵 = (
2 1 3 −2
0 −2 2 4
1 1 3 2
0 1 −1 1
)
21. Find the inverse of the matrices 𝑋 = (
1 0 0
1 1 1
0 0 1
) and 𝑌 = (
2 0 1 0
1 1 2 1
0 1 0 1
1 0 1 2
).
22. Let 𝐴𝐵 = (
2 1
10 1
) and 𝐵 = (
4 1
1 1
). Find 𝐴.
23. Let 𝐴 and 𝐵 be two invertible matrices. Show that (𝐴𝐵) 𝑇
= 𝐴 𝑇
𝐵 𝑇
if and only if (𝐴𝐵)−1
= 𝐴−1
𝐵−1
.
24. Show that 𝐴2
is invertible and (𝐴2)−1
= 𝐵 if and only if 𝐴 is invertible and 𝐴−1
= 𝐴𝐵.
25. A square matrix 𝐴 is said to be orthogonal if 𝐴 𝑇
= 𝐴−1
.
25.1. Show that if 𝐴 and 𝐵 are orthogonal, then 𝐴𝐵, 𝐴𝐵 and 𝐴 𝑇
are orthogonal.
25.2. Show that the rotation matrix of the exercise 13. is orthogonal.

3. 3
Miguel Fernandes
26. Give examples of non-invertible matrices 𝐴 and 𝐵 such that 𝐴 + 𝐵 is invertible.
27. Let 𝐴 and 𝐵 be square matrices. Show that if 𝐼 − 𝐴𝐵 is invertible, then 𝐼 − 𝐵𝐴 is also invertible.
Hint: Consider 𝐵(𝐼 − 𝐴𝐵)−1
𝐴 + 𝐼.
28. Compute the determinant of the following matrices.
28.1. 𝐴 = (
21 3
1 2
)
28.2. 𝐵 = (
2 2 5
1 1 4
3 5 9
)
28.3. 𝐶 = (
2 0 0 0
0 1 2 1
0 2 0 1
0 0 1 2
)
29. Given |
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
| = 2, find the following determinants.
29.1. |
𝑎 𝑏 𝑐
2𝑑 2𝑒 2𝑓
𝑔 ℎ 𝑖
|
29.2. |
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 + 𝑑 ℎ + 𝑒 𝑖 + 𝑓
|
29.3. |
𝑑 𝑒 𝑓
𝑎 𝑏 𝑐
𝑔 ℎ 𝑖
|
30. Show that |
𝑏 + 𝑐 𝑎 𝑎
𝑏 𝑐 + 𝑎 𝑏
𝑐 𝑐 𝑎 + 𝑏
| = 4𝑎𝑐.
31. Consider a 3-square matrix 𝐴 such that det 𝐴 = 2. Find:
31.1. det(−2𝐴).
31.2. det 𝐴3
.
31.3. det 𝐴−2
.
32. Solve the following equation:
|
𝑥 0 0
0 1 0
1 1 𝑥
| = 1
33. Let 𝐴 be a (real) square matrix. Mark each of the following statements True or False. Justify your answer.
33.1. det 𝐴𝐴 𝑇
is a non-negative number.
33.2. If 𝐴 𝑛
= 0, for some 𝑛 ∈ ℕ, then det 𝐴 = 0.
33.3. If 𝐴 is orthogonal (exercise 25.), then det 𝐴 = 1.
33.4. If 𝐴 is an idempotent matrix, 𝐼 − 𝐴 is also idempotent.
33.5. If 𝐴 is a skew-symmetric matrix of order 𝑛, 𝑛 an odd number, then det 𝐴 ≠ 0.
33.6. If 𝐴 is a skew-symmetric matrix of order 𝑛, 𝑛 an even number, then nothing can be concluded about the
determinant of 𝐴.
33.7. If det 𝐴2
− 2 det 𝐴 + 1 = 0, then 𝐴 is invertible.
34. Find all invertible matrices of order 2 such that det 𝐴2
= det(−2𝐴3
).