MAT1503 ASSIGNMENT 4 ANWERS 2023

Total Marks: 100

ONLY FOR YEAR MODULE This assignment covers chapter 2 & 3 of the prescribed book as well as the
study guide, it is based on Study Units 2.1, 2.2, 2.3 & 3.1



Question 1

(1.1) Let's find the matrix E1:



E1 =

[1 0 0]

[0 1 -2]

[0 0 1]



Performing the operation (3) R2 ←− R2 − 2R1, we have:



E1A =

[1 0 0]

[0 1 -2]

[0 0 1]

*

[2 -1 1]

[3 1 -1]

[1 -3 k]

,=

[2 -1 1]

[3 1 -1 - 2(2 -1 1)]

[1 -3 k]



=

[2 -1 1]

[-1 3 -3 -2 -1 2]

[1 -3 k]



=

[2 -1 1]

[-1 3 -3 -2 -1 2]

[1 -3 k]



=B



Now, let's find the values of k for which a33 = a23:



a33 = a23

1 = -3

This is not possible, so there are no values of k in the range [-1, 0] that satisfy the condition.



(1.2) Let's find the matrix E2:



E2 =

[0 0 1]

[0 1 0]

[1 0 0]



Performing the operation (3) R1 ←→ R3, we have:

, A=

[0 0 1]

[0 1 0]

[1 0 0]

*

[2 -1 1]

[3 1 -1]

[1 -3 k]



=

[1 -3 k]

[3 1 -1]

[2 -1 1]



=C



So, the matrices E2 and C are:



E2 =

[0 0 1]

[0 1 0]

[1 0 0]



C=

[1 -3 k]

[3 1 -1]

[2 -1 1]



Question 2