Download Course: Engineering Mathematics I (MATH 1180) Group I: CIV

Course: Engineering Mathematics I (MATH 1180)
Assignment #3: MATRICES
MATH 1180: 
Group I: CIV, ELEC, SURV.
Assignment #4: MATRICES
Lecturer: Mr. Oral Robertson
Date: November 4, 2010
Due on: Wednesday, November 10, 2010.
*THIS PAPER HAS 2 PAGES and 9 QUESTIONS.
1. (a) It is known that matrix multiplication is not commutative, but is associative. Given the following
 1 2 
 0 1
3 4
 2 3
 , Q  
 , R  
 , and S  
 ,
matrices: P  
 1 3
 1 2 
2 0
 4 1
Verify that: (PQ)RS = P(QR)S.
(b) By applying the associative property of matrix multiplication on the product B –1 A–1(AB), or
otherwise, prove that: (AB) –1 = B –1 A–1.
(c) Using the identity in part (b) above, prove that: (ABC) –1 = C–1 B–1 A–1 .
b
a
 , where a, b, c, and d have integral values, such
2. (a) Find the four (4) matrices of the form A  
d 
c
that the matrices have a determinant of 1 and are also orthogonal.
 x 1
 2 0
 such that AT + |A|A–1 = 
 .
(b) Find, if possible, all matrices of the form A  
 y z
 0 2
3. (a) Prove that:
(i) (AB)T = BTAT …….. from first principles.
(ii) (A–1)T = (AT) –1  Hint: Consider the product AT(A–1)T .
(iii) A3 = A  A = A–1.
(b) Prove the following statements (assuming square-, multipliable, and/or invertible- matrices):
(i) If A and B commute (that is, AB = BA), and A and B are both symmetric, then AB is symmetric.
(ii) If A and B are orthogonal, then AB is orthogonal.
(iii) If A and B are symmetric, then AB – BA is skew-symmetric.
(iv) If A is an m-square symmetric matrix and P is of order m×n, then PTAP is symmetric.
(v) If A is an m-square skew-symmetric matrix and P is of order m×n, then PTAP is skew-symmetric.
4. By appropriate successive pre-multiplication and/or post-multiplication of the given equations by a
matrix (or its inverse), obtain the following results from the given equations.
(i) Given that AB–1C = D, then C = BA–1D and A = DC–1B.
(ii) Given that A + B = AB, then A–1 + B–1 = I.
(iii) Given that A + B = BC, then A–1 + B–1 = CA–1.
(iv) If A–1 + B = A + B–1, then: (A – B) = A(B – A)B.
2

5. (a) (i) If A   4
1

3
3
2
4
4
  10


1  and B   15  4
 5 1
4 

9

 14  , show that: AB = 5I.
6 
Hence, solve the system of equations: –10x + 4y + 9z = 10, 15x – 4y – 14z = 20, –5x + y + 6z = –30.
(ii) Express matrix A in part(i) above as the sum of a symmetric and a skew-symmetric matrix.
 Hint: if A is a square matrix, then A + AT is symmetric, while A – AT is skew-symmetric.
(b) Find the rank of each of the following three matrices: [Use row-by-row reduction, if necessary].
1 2 3 


A   2 3 4 ,
3 5 7 


1

2
B
4

2

1
1
4
1
1
2
4
2
6

6
,
9

7 
4
 1 2 1

C 2 4 3
5
1  2 6  7



.


1
Course: Engineering Mathematics I (MATH 1180)
Assignment #3: MATRICES
Group I: CIV, ELEC, SURV.
Lecturer: Mr. Oral Robertson
Date: November 4, 2010
6. (a) Use Row-by-Row elimination to show that following system of equations has infinite solutions.
6x + 3y + 2z = 6
10x + 5y + 6z = 10
2x + y – 3z = 2 .
Obtain the solution to the system (in terms of one or more free variables, as appropriate).
(b) In trying to solve a system of equations, the final row-reduced form of the augmented matrix was
as follows:
1

0
0

2
3
c
d
0
a
1 

e 
b 
Choose any set of values for the constants a, b, c, d, and e that would make the system:
(i)
(ii)
(iii)
(iv)
Have unique solutions
Have infinite solutions with one free variable
Have infinite solutions with two free variables
Be inconsistent.
7. Find the values of the rational numbers a and b that makes the following system
x – 2y + 3z = 4
2x – 3y + az = 5
3x – 3y + 5z = b .
has:
(i ) unique solution,
(ii) infinite solutions, (iii) no solution.
Find the unique solution corresponding to the values a = 3 and b = 0.
Find the set of infinite solutions.
8. Find the values of the real scalars m and n for which the set of equations
x + 3y + 2z
(m–1)y + 2z
2x + my + (m–2)z
has:
(i) unique solution,
= 9
= 2
= n–2 .
(ii) infinite solutions, (iii) no solution (inconsistency).
Find the unique solution corresponding to the values m = 0 and n = 0.
Find the set of infinite solutions corresponding to the highest value of m.
9. Define the following terms:
(i)
column matrix
(ii)
square matrix
(iii)
singular matrix
(iv)
adjoint of a [square] matrix A
(v)
rank of a matrix
(vi)
transpose of a matrix A
(vii) leading diagonal
(viii) symmetric matrix
(ix)
skew-symmetric matrix
(x)
upper-triangular matrix
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