Linear Algebra Answers

3. If the matrices A and B are given by A= and B =

a. Calculate 3A -2B+7I

b. 3A + 4X = B is an equation for an unknown matrix X. Find the matrix X

c. Calculate the matrix product AB

4. Solve the following set of equations by Gaussian elimination:

3x + 2y + z =2

4x + 2y + 2z =8

x – y + z = 8

Define T:R^3 -R^3 by

T(x1,x2,x3) = (x1+x2,x2+x3,x1-x3,2x1+x2-x3)

Check that T is a linear operator . Find the kernal and range of T. Find the dimension of the kernal.

Given:

A=

4 0 0

2 2 3

1 1 1

B=

4 0 0

2 2 3

1 1 1


Find: a) –A-1+ 3BT b) B-1+ ( AT+A-1)

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.