MATH 217 Linear Algebra
Worksheet 1
September 23, 2018
1. For the matrix A and the vector (column- (a) Find, if possible,
matrix) v given below, compute the product
Av. A+B, B +C, 5A, AB, BC, CB, A2 , C 2 .
2 5 5 (b) Find a matrix D such that
(a) A = and v = .
−3 −1 6
A − 3B + D = BC 2 + 2A.
1 −3
5
(b) A = −2 1 and v = . (c) Find B T , C T , (BC)T .
−5
3 −1
3 −1
6 2 7 5. Let A = . If A2 − 2A − 8I2 = 02
(c) A = and v = . −5 a
−3 −1 −1
determine a.
5 −3 1 0
2 −1 + i
(d) A = −2 1 4 and v = 22 .
6. If v = 3i and w = 2 find v T w
1 0 −2 −11
1−i 3−i
T
and w w.
2 1
2. Consider the square matrices A =
−1 3
x 1
−1 3
7. If A = , determine all values of x and
and B = . −2 y
0 4 y for which A2 = A.
(a) Is it true that AB = BA ?
2 2 1 −x −y z
(b) Compute A2 and B 2 . 8. Let A = 2 5 2, B = 0 y 2z .
1 2 2 x −y z
3. Let A, B be square matrices of the same size.
Then (A + B)2 may not be equal to thematrix Find all values x, y, z such that
2 2 1 2
1 0 0
A + 2AB + B . Observe this for A = ,
0 0 B T AB = 0 1 0 .
1 0 0 0 7
B= .
3 0
4. For the matrices 9. The commutator of two square matrices A and
B is defined as [A, B] = AB − BA. Compute
3 −2 1 1 5 1 [A, B] for the following matrices:
A= , B = and,
−1 0 5 2 3 −1
3 0 2 5
(a) A = ,B= .
0 3 −7 4
2 −2 3
C = −1
1 1
1 i 0
1 0 i
3 1 1 (b) A = ,B= .
2 0 −i 2 i 0
answer the following questions:
1
Worksheet 1
September 23, 2018
1. For the matrix A and the vector (column- (a) Find, if possible,
matrix) v given below, compute the product
Av. A+B, B +C, 5A, AB, BC, CB, A2 , C 2 .
2 5 5 (b) Find a matrix D such that
(a) A = and v = .
−3 −1 6
A − 3B + D = BC 2 + 2A.
1 −3
5
(b) A = −2 1 and v = . (c) Find B T , C T , (BC)T .
−5
3 −1
3 −1
6 2 7 5. Let A = . If A2 − 2A − 8I2 = 02
(c) A = and v = . −5 a
−3 −1 −1
determine a.
5 −3 1 0
2 −1 + i
(d) A = −2 1 4 and v = 22 .
6. If v = 3i and w = 2 find v T w
1 0 −2 −11
1−i 3−i
T
and w w.
2 1
2. Consider the square matrices A =
−1 3
x 1
−1 3
7. If A = , determine all values of x and
and B = . −2 y
0 4 y for which A2 = A.
(a) Is it true that AB = BA ?
2 2 1 −x −y z
(b) Compute A2 and B 2 . 8. Let A = 2 5 2, B = 0 y 2z .
1 2 2 x −y z
3. Let A, B be square matrices of the same size.
Then (A + B)2 may not be equal to thematrix Find all values x, y, z such that
2 2 1 2
1 0 0
A + 2AB + B . Observe this for A = ,
0 0 B T AB = 0 1 0 .
1 0 0 0 7
B= .
3 0
4. For the matrices 9. The commutator of two square matrices A and
B is defined as [A, B] = AB − BA. Compute
3 −2 1 1 5 1 [A, B] for the following matrices:
A= , B = and,
−1 0 5 2 3 −1
3 0 2 5
(a) A = ,B= .
0 3 −7 4
2 −2 3
C = −1
1 1
1 i 0
1 0 i
3 1 1 (b) A = ,B= .
2 0 −i 2 i 0
answer the following questions:
1
