Linear Algebra Answers

Solve the following system of linear equations by using the inverse matrix
method: (10)
x + y + z = 4
-2x - y + 3z = 1
y + 5z = 9

Let A and B be any two matrices such that B is the inverse of A.
3.1 Determine the relationship between the adjoint of A and the adjoint of B. (5)
3.2 Determine the relationship between the transpose of A and the transpose of B.

Consider the linear system:

x + 2y + 3z = a
x + 3y + 8z = b
x + 2y + 2z = c

where a, b and c are arbitrary constants. Find all solutions of this system.

(12)

Please assist.

Explain the difference between a singular and a non-singular matrix. Show that a non-singular matrix
must be square.

Is the matrix
A =


1 1 1
2 3 0
3 8 3

 .
invertible? (5)
4.2 Solve the following system by Gauss-Jordan elimination
x + y + z = 2
2x + 3y + 2z = 5
3x + 8y + z = 11
(10)
4.3 Now solve the system in 4.2 by applying Cramer’s Rule.

Let a11 x1 + a12 x2 + a13x3 = b1
a21 x1 + a22 x2 + a23 x3 = b2
a31 x1 + a32 x2 + a33 x3 = b3.
Show that if det (A) 6= 0 where det(A) is the determinant of the coefficient matrix,
then x2 =
det(A2)
det(A) where det(A2) is the determinant obtained by replacing the second column of det(A)
by (b1, b2, b3)
T

Suppose
A =


4 3 5
1 3 −5
2 1 5

 .
6.1 Evaluate det (A) by expanding along the 2nd row. NO marks will be awarded if you use any other
method.
(5)
6.2 Can Cramer’s rule be used to solve the system
A


x
y
z

 =


0
0
a

 where A is given and a 6= 0?
Give reasons for your answer. If the answer is Yes, use Cramer’s rule to solve the system. If not, use
any other method.

State De Moivre’s Theorem. (2)
9.2 Express cos 5θ and sin 4θ as polynomials in terms of sin θ and cos θ. (8)
9.3 Let w be a negative real number, z a 6
th root of w.
(a) Show that z (k) = ρ
1
6
-

cos

Consider the linear system
x + 2y + 3z = a
x + 3y + 8z = b
x + 2y + 2z = c



where a, b and c are arbitrary constants. Find all solutions of this system.

Consider the following homogeneous system of linear equations
x + 4y + z = 0
4x + 13y + 7z = 0
7x + 22y + 3z = 0



.
(a) Determine the solution(s) of the above system by reducing the system to row echelon
form. (7)
(b) Is (2, 4, 2) a solution of the system? Give a reason for your answer. (3)
(c) State how you can use the determinant of the coefficient matrix of the above system to determine
if the system has nontrivial solutions or not. DO NOT EVALUATE THE ACTUAL DETER-
MINANT.