Finite Mathematics and Linear Algebra 102 Problem sheet 11 ... - DCU

Finite Mathematics and Linear Algebra 102
Problem sheet 11
⎡ ⎤
⎡ ⎤ ⎡ ⎤
3 0 [ ] [ ] 1 5 2
6 1 3
1. A = ⎣−1 2⎦ 4 −1 1 4 2
, B = , C = , D = ⎣−1 0 1⎦ , E = ⎣−1 1 2⎦ .
0 2 3 1 5
1 3
3 2 4
4 1 3
Compute the following where possible
(a) 5A , 2B − C, 4T r(7B), T r(A).
(b) 1 2 CT − 1 4 A, 1 3 E − 2D.
(c) AB, BA, (AB)C, A(BC), DE, C T BA T .
(d) (4B)C + 2B, B T (CC T − A T A).
2. Using the matrices in Exercise 1 calculate the following:
(AB) T , A T B T , B T A T , (AC) T , C T A T , (DE) T , E T D T , (BC) T , C T B T .
What conjecture can you make?
3. Indicate whether the statement is always true or sometimes false. Justify your answer with
a logical argument or a counter example. [Hint: in each case work out five examples, or try
to find a counter example]
(a) The expressions T r(AA T ) and T r(A T A) are always defined regardless of the size of A.
(b) T r(AA T ) = T r(A T A) for every matrix A.
(c) If the first column of A has all zeros, then so does the first column of every product AB.
(d) If the first row of A has all zeros, then so does the first row of every product AB.
[ ]
[ ]
a b
d −b
4. Let A = , | ad − bc ≠ 0. Prove that A
c d
−1 = 1
.
ad−bc −c a
5. A square matrix is called symmetric if A T = A, and skew-symmetric if A T = −A.
(a) If A is a square matrix show that: AA T and A + A T are symmetric. Show also A − A T
is skew-symmetric.
(b) Determine all 2 × 2 symmetric matrices.
(c) Determine all 2 × 2 skew-symmetric matrices.
≠
6. Let A, B be square matrices of the same size. Give an example in which (A + B) 2
A 2 + 2AB + B 2 .
1

7. A matrix A is said to be an involution is A 2 = I. Note A 2 = A · A.
[ ] 1 −2
(a) Show that is an involution.
0 −1
[ ] 5 3
(b) Find the values of a and b for which the matrix is an involution.
a b
(c) Find six different 2 × 2 matrices which are involutions and are skew-symmetric.
(d) Suppose A is an n × n matrix which is also an involution. What is A −1 .
8. Let A be an n × n matrix. Let A n be the product A · A · A.. · A, the product A with itself n
times. Show that (A n ) −1 = (A −1 ) n .
2