Course 111: Algebra, 14th Feb 2007
To be handed in at tutorials on Feb 19th and 20th.
1. Solve the following system of linear equations using Gauss-Jordan elimination.
2x1 + x2 − x3 = 2
x1 − x 2 + x 3 = 7
2x1 + 2x2 + x1 = 4
What is the rank of the coefficient matrix describing this system?
Write the augmented matrix, Ag and perform row operations to transform it to reduced row-echelon form:


2 1 −1 2

Ag =  1 −1 1 7 

2 2
1 4
e.r.o0 s
→


1 0 0 3


 0 1 0 −2 
0 0 1 2
Where the row operations were 21 R1 , R2 − R1 , R3 − 2R1 , − 32 R2 , R3 −
R2 , 13 R3 , R2 + R3 , R1 + 12 R3 , R1 − 12 R2 . Giving solution: x1 = 3, x2 =
2, x3 = 2.
The rank of A is 3.
2. Compute the inverses of the following matrices
−1

3 0 0


 0 2 0 
9 5 4
Using the row operations:


0
0
0 


1
= 0
2
− 34 − 58
1
R , 1 R , R3
3 1 2 2
−1

1
3
1 −2 0


 −2 3 1 
0
1 0
(1)
1
4
− 9R1 , R3 − 5R2 , 41 R3 .


1 0 2


= 0 0 1 
2 1 1
(2)
Using the row operations: R2 + 2R1 , R2 ↔ R3 , R1 + 2R2 , R3 + R1 .

−1
1 −1 3

1 2 
 2

−2 −2 1
− 31

2
=  − 15

2
5
2
3
1
15
− 15

0
0 

0)
(3)
Using the row operations: R2 −2R1 , R3 +2R1 , R3 −4R1 , 31 R2 , − 15 R3 , R1 +
R2 , R1 − 53 R3 , R2 + 34 R3 .
Given the matrix


−2 0 −4 1


D =  1 0 −1 6 
8 2 1 −1
Determine D T D.




DT = 
−2 1
8
0
0
2
−4 −1 1
1
6 −1





(4)
Therefore the product D T D is




DT D = 
−2 1
8
0
0
2
−4 −1 1
1
6 −1

69 16 15 −4
−2 0 −4 1

 16
4
2
−2



  1 0 −1 6  = 

 15
2
18 −11
8 2 1 −1
−4 −2 −11 38
(5)








Note that the product is symmetric even though D was not symmetric
(or even square!).