MH1810 Tut 3 2018 Matrices
MH1810 Tut 3 2018 Matrices
MH1810 Tut 3 2018 Matrices
1. Find matrices A = [aij ], B = [bij ] and C = [cij ] whose sizes and entries satisfy the stated conditions.
8
< 1 if i < j;
(a) A is 3 4 and aij = 7 if i = j;
:
9 if i > j:
1 if ji jj > 1;
(b) B is 4 4 and bij =
1 if ji jj 1:
1 i = j; if
(c) C is 3 2 and Cij =
2i 3j i 6= j:if
2 3
2 3 1 1 1 1 2 3
7 1 1 1 6 1 1 18
1 1 1 7
(Answers: A = 4 9 7 1 1 5,B=6
4 1
7, C = 4 12 1 5 .)
1 1 1 5
9 9 7 1 24 72
1 1 1 1
2. Consider the matrices
2 3 2 3 2 3
3 0 1 5 2 6 1 3
4 1 1 4 2
A = 4 1 2 5; B = ; C= ; D=4 1 0 1 5; E = 4 1 1 2 5
0 2 3 1 5
1 1 3 2 4 4 1 3
Compute the following (where possible). Note M T means the transpose of M . It is obtained
2 from M
3
1 3
1 4 2
by interchanging the rows and columns. Example: For M = , we have M T = 4 4 1 5.
3 1 5
2 5
(a) BA (b) BC (c) DT ET (d) (D E)T (e) DE (f) (DA)T
(Answers: (a) Not de…ned; 2 3 2 3
5 0 1 9 8 19
1 15 3 0 2 11
(b) ; (c) & (d) 4 4 1 1 5; (e) 4 2 0 0 5; (f) .)
6 2 10 12 1 8
1 1 1 32 9 25
3. Let 2 3 2 3
3 2 7 6 2 4
A=4 6 5 4 5 and B = 4 0 1 3 5
0 4 9 7 7 5
(a) If A is a square matrix with two identical rows, then AA has two identical rows.
(b) If A is a square matrix and AA has a column of zeros, then A must have a column of zeros.
(c) If the matrix sum AB + BA is de…ned, then A and B must be square.
3 2
5. Let A = .
2 3
6. Find A where
1 3 7 1 3 1
(a) (7A) = (b) (I + 2A) =
1 2 1 2
5 1
1 2 7 14 14
(Answers: (a) A = 7 ; (b) A = 1 4 )
1 3 14 14
2 k 2
7. Consider the matrix A where A = .
3 k
(c) State the value of k for which there are no solutions to the simultaneous solutions. Justify your
answer.
k (k 2) x 10 k
(Answers: (a) k 6= 6; (b) A 1 = 6 1 k , = 6 1k (c) k = 6 )
3 2 y 2
8. Suppose the …rst row of an n n matrix A is identical to the second row of A. Is there an n n matrix
B such that AB = I? Is A invertible?
9. The trace of a square matrix is de…ned to be the sum of its diagonal entries. We denote by tr(A) the
trace of a square matrix A.
This year
Mass transit Automobile
Next Mass transit 0:7 0:2
year Automobile 0:3 0:8
For example, 30% of commuters taking mass transit this year will change to driving automobile next
year.
Suppose the population of the area remains constant, and that initially 30 percent of the commutes
use mass transit and 70 percent use their automobiles.
What percentage of commuters will be using the mass transit system after 1 year? After 2 years?
(Answers: 35% and 37.5% )
Challenging Question