Linear Algebra Exercices List 1: Matrix Algebra
1. Write as an array:
(a) A:{1,2} × {1,2} 7→R (i, j)yi+j (b) B :{1,2} × {1,2} 7→R (i, j)yi·j
(c) C :{1,2,3} × {1,2,3} 7→R (i, j)y
(1 if i=j 0 if i6=j
2. Determine the additive inverse −A of A =
1 0 −1
2 √
2 π
. 3. Let
A=
1 0 2 1
B =
−1−2 0 1
Determine −A+ 5B.
4. Let A, B ∈ Mn×n and k ∈R. Answer the following questions:
• IfA, Bare diagonal matrices, areA+BandkAdiagonal matrices?
• If A, B are upper triangular matrices, are A+B and kA upper triangular matrices?
• IfA, B are symmetric matrices, are A+B andkAsymmetric ma- trices?
• IfA andB are skew symmetric matrices, are A+B and kAskew symmetric matrices?
5. Write
1 2 0
1 2 −1
0 1 −2
as a sum of a symmetric matrix with a skew symmetric matrix.
6. Show that, if a matrix A ∈ Mn(R) is both symmetric and skew- symmetric, then A= 0n×n. Conclude that the decomposition
A=S+T
where S is symmetric and T skew symmeric is unique. I.e, if A=S1+T1 =S2+T2
whereA∈ Mn(R) andS1, S2 are symmetric andT1, T2skewsymmetric, then
S1 =S2 and T1 =T2
7. Consider the spatial vectorsu= (1,0,1) ev = (0,1,0). Verifyu·v = 0.
Try to represent them. What do you observe concerning the angle between them?
8. CalculateAB and BA whereA and B were given in Exercice 1. Same demand with BC and CB. Is matrix multiplication commutative?
9. Write as matrix multiplication the following system
x−z =−2 x+z = 4 x+y+z= 6 (and why not try to solve it?)
10. Consider the set C of 2×2 matrices of type:
a −b b a
where a, bare real numbers. Verify the following properties:
(a) I ∈ C.
(b) IfA, B ∈ C then A+B ∈ C.
(c) IfA, B ∈ C then AB=BA∈ C.
(Do you find any resemblance between C and the set of complex numbers C?)
11. (a) Let
A=
0 1 0 0
Calculate A2. Which matrix stands for An whenn ≥2?
(b) Let
B =
0 −1
1 0
Verify that B2 =−I. Determine B2021.
12. (adapted from “Linear Algebra”, Lipschutz and Lipson) Let
A=
1 0 2
2 −1 3
4 1 8
and B =
−11 2 2
−4 0 1
6 −1 −1
(a) Show thatB =A−1. (b) Compute the inverse of
C = 2·
1 2 4
0 −1 1
2 3 8
(hint: note that C = 2·AT) (c) Solve (quickly) the system
A·
x y z
=
1 0 0
13. Compute, if possible, the inverses of the following matrices:
(a)
3 1 2 4
(b)
6 9
−2 −3
(c)
2 0 0
0 −1 0
0 0 √
2
(d)
1 0 0
0 0 3
0 1 8
Hints or solutions:
Exercice 1 (a):
2 3 3 4
; (b)
1 2 2 4
Exercice 3:
−6 −10
−2 4
Exercice 4: All answers are positive.
Exercice 8AB =
8 16 11 22
and BA=
8 11 16 22
Exercice 11: B2021 =B.
Exercice 12.
(1) Just verify thatAB =BA =I3. (2) Note that
C−1 = (2AT)−1 = 1
2 ·(AT)−1 = 1
2 ·(A−1)T = 1 2·BT
(3) Multiply both sides of the equality by A−1 = B to determine the solution.
Exercice 13.
(1)-(2) Use proposition 1.6.
(3) Note that ofA is diagonal and invertible, A−1 is also diagonal.
(4)
1 0 0
0 −8/3 1
0 1/3 0
.
Test and exam exercices:
1. Consider the matrices A=
1 1 2
0 1 −1
, B = 1
2 1
2 0
0 1 −1
Compute C = (−A) + 2B and D=A·(BT).
2. Consider the matrix
A=
1 1 1
0 1 −1
0 0 −1
Explicit −2A, AT, A+AT and A−AT.
3. Consider the matrix
A=
a b a
a−b 0 1 b+ 1 0 1
Determine all pairs a and b such thatA is symmetric.
4. Consider the matrices A=
−1 2 5
1 0 1
and B =
1 0
−2 δ
−δ 0
.
Determine, if possible, a value for δ such thatAB is skew-symmetric.
5. Compute the inverse of the matrix
0 1 0
3 0 0
0 1 1
6. Let
A =
1 −1 2
0 1 1
1 −1 1
and B =
−2 1 3
−1 1 1
1 0 −1
(a) Compute A·B and conclude about the invertibility of A.
(b) Solve the linear system
1 −1 2
0 1 1
1 −1 1
·
x y z
=
1 0 1
(Hint: the system can be quickly solved once you know the inverse of A)
Supplementary exercices:
Solved problems 2.1-2.30, Linear Algebra, Lispchutz and Lipson, sixth edition.
