Solution:

e)

d)

c)

b)

a)

then , the third row and second column of C equals

If C = AB where

10.2.1. (001-Final-5 $10.2)

Section 10.2

Thus,

Solve the system for a, b and c.

Note: We only need to compute the first row of the product and equate it to the first row of the identity matrix.

Solution:

e)

d)

c)

b)

a)

The sum x + y +z equals:

The matrix M and its inverse are given by

10.3.1 (001-Final-18 $10.3)

Section 10.3

is obtained by multiplying the 3rd row of A by the second column of B.

and

or

==>

and

Case III (No solution)

==>

and

or

Substitute the given point into the equation of the parabola to get a system of 3 equations in 3 unknowns.

Solution:

e)

d)

c)

b)

a)

10.1.6 (002-Final-17 $10.1)

==>

Solution:

e)

d)

c)

b)

a)

10.4.2 (002-Final-18 $10.4)

==>

==>

2R1+R4

----------->

Solution:

e)

Solution:

e)

d)

c)

b)

a)

10.4.3 (002-Final-25 $10.4)

Add the solutions.

Solve for x.

Compute the determinant.

d)

c)

b)

a)

10.3.2 002-Final-16 $10.3

==>

==>

==>

==>

d)

c)

b)

a)

What is the value of the following determinant?

10.4.1 (001-Final-8 $10.4)

Section 10.4

Thus,

Compute the reduce row echelon form of the augmented matrix to find

Augment the matrix A with the identity matrix

Solution:

e)

Mathcad check:

Use the rref command to compute the reduced row echelon form.

and

Thus,

This is the reduced row echelon form.

5R3+R1

==>

-3R3+R2

==>

This is the echelon form. We can now use back substitution to find x, y and z.

However, it is better to complete the reduction to find the reduced row echelon form and then read the solution from the reduce matrix as shown below.

-R2+R1

==>

-4R2+R3

then the system becomes

and

Let

Solution:

e)

d)

c)

b)

a)

10.1.3 (001-Final-26 $10.1)

==>

-2R1+R2

Solution:

e) The system is consistent for c > 0.

d) The system is inconsistent for all values of c.

c) The system can be made consistent for a suitable choice of c.

b) The system is consistent for all , with exactly one solution.

a) The system is consistent if c = 0, with infinitely many solutions.

Which one of the following statements is TRUE about the linear system of equations which has the augmented matrix

10.1.1 (001-Final-12 $10.1)

Section 10.1

A. M. Farhat

Solution of Old Exams Problems

Chapter 10

Math002

R1 <==> R3

-R1+R2

==>

-4R1+R3

Solution:

e)

d)

c)

b)

a)

10.1.2 (001-Final-22 $10.1)

The system has a leading entry in the last column ==> the system has no solution regardless of the choice of c.

-R1+R3

==>

d) and

c) and

b)

a) and

10.1.5 (002-Final-15 $10.1)

Not equal

(c)

Not equal

(b)

==>

and

Case II (Infinite number of solutions)

or

==>

Case I (Unique solution)

Reduce the augmented matrix.

-2R1+R2

Solution:

e)

==>

-2R3+R2

==>

-4R3+R1

==>

R2+R1

==>

-5R2+R3

R2<==>R3

-3R1+R2 ==>

Augmented matrix of the new system

(a)

Examples:

10.1.4 (002-Final-7 $10.2 )

==>

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