CEGEP CHAMPLAIN - ST.
LAWRENCE Problem Sheet #4
201-105-RE: Linear Algebra
Patrice Camiré
The Trace and Transpose of a Matrix
1. Let A = [aij ]n×n . We denote and define the trace of A by tr(A) = a11 + a22 + · · · + ann .
In other words, the trace of A is the sum of its entries along the main diagonal.
(a) Compute the trace of A if
1 3 1 0 3 1 2 −1 0
i. A =
4 −6 ii. A = 2 3 2 0 3 4 0
iii. A =
0 1 4 7 4 −5 8
6 0 2 1
(b) Show that tr(kA) = k · tr(A), for any k ∈ R.
(c) Show that tr(A ± B) = tr(A) ± tr(B), for any A, B ∈ Rn×n .
(d) (Optional) Show that tr(AB) = tr(BA), for any A ∈ Rm×n , B ∈ Rn×m .
Why is this not obvious?
(e) Show that it is impossible to find matrices A, B ∈ Rn×n such that AB − BA = In , where In
is the n × n identity matrix. (Hint: Use the trace.)
(f) Describe explicitly all 2 × 2 matrices with zero trace. Write your generic matrix as a linear
combination of three matrices. (There are two possible natural answers in this case.)
2. Let A ∈ Rm×n . The transpose of A is denoted by AT ∈ Rn×m and is defined by the equality
(AT )ij = (A)ji . In other words, the transpose of A is obtained by interchanging the rows and
columns of A. Compute the transpose of A in each case, that is, find AT .
1 2 6 −3 2
(a) A =
−4 5 (d) A = 1 5 3
4 9 7
8 4 3 −1
(b) A =
0 −2 5 9
1 (e) A = 5 −4 −3
0
(c) A = 1
1 2 3 −5
0 (f) A = 0 1 −8 9
6 −6 6 5 0
3. (Optional) Prove the following properties of the transpose. Assume that addition/multiplication is
defined where applicable.
(a) (AT )T = A.
(b) (kA)T = kAT , for any k ∈ R.
(c) (A ± B)T = AT ± B T .
(d) (AB)T = B T AT .
(e) (A1 A2 · · · Am )T = ATm · · · AT2 AT1 . (Apply (d) recursively.)
(f) (An )T = (AT )n , for any n ∈ N. (Apply (e) directly.)
4. A square matrix A is said to be symmetric if AT = A and skew-symmetric if AT = −A.
(a) Given a square matrix A, show that the following matrices are symmetric: AAT , AT A and
A + AT .
(b) Describe all 2 × 2 symmetric matrices explicitly and express your generic symmetric matrix as
a linear combination of three matrices.
(c) Given a square matrix A, show that A − AT is skew-symmetric.
(d) Describe all 3 × 3 skew-symmetric matrices explicitly and express your generic skew-symmetric
matrix as a linear combination of three matrices.
(e) Show that any square matrix A can be expressed as the sum of a symmetric and a skew-
symmetric matrix. (Hint: Consider A + AT and A − AT .)
Answers
1. (a) i. tr(A) = 1 − 6 = −5 ii. tr(A) = 1 + 3 + 4 = 8 iii. tr(A) = 1 + 3 − 5 + 1 = 0
(b)
(c)
(d) Use sigma notation and sum the terms in a different order.
This is not obvious since AB and BA may not be of the same size. Furthermore, even if
AB and BA are of the same size, we know that AB 6= BA in general.
(e) Use the trace and its properties stated above to prove the result.
a b 1 0 0 1 0 0
(f) =a +b +c
c −a 0 −1 0 0 1 0
1 −4 (c) AT =
T 1 0 1 0 6 5
2. (a) A =
2 5 (e) AT = −4
−3
8 0 1 0 −6
4 −2 6 1 4 2 1 6
(b) AT = (f) AT =
(d) AT = −3 5 9
3 5 3 −8 5
−1 9 2 3 7 −5 9 0
3.
4. (a) Show this directly using properties of the transpose.
(b) All 2 × 2 symmetric matrices are of the form
a b 1 0 0 1 0 0
=a +b +d
b d 0 0 1 0 0 1
(c) Show this directly using properties of the transpose.
(d) All 3 × 3 skew-symmetric matrices are of the form
0 a b 0 1 0 0 0 1 0 0 0
−a 0 c = a −1 0 0 + b 0 0 0 + c 0 0 1
−b −c 0 0 0 0 −1 0 0 0 −1 0
(e) It’s almost enough to add up the two matrices provided in the hint.
