Mathematics IAWorked Examples

ALGEBRA: MATRICES AND LINEAREQUATIONS

Produced by the Maths Learning Centre,The University of Adelaide.

May 1, 2013

The questions on this page have worked solutions and links to videos onthe following pages. Click on the link with each question to go straight tothe relevant page.

Questions

1. See Page 4 for worked solutions.

Let A =

[0 1 −1−1

20 1

2

], B =

305

, C =[1 −1 1

]and D =

[10 05 2.5

].

Also let I be the 3 × 3 identity.For each of the following, state if the operation is defined and calculatethe result if it is defined.

(a) BA (b) AB (c) AtD (d) DAt

(e) BC (f) CB (g) B + C (h) (Bt + C)(BC + I)

2. See Page 7 for worked solutions.Show that if A, B are (any) two matrices such that AB and BA are bothdefined then AB and BA are both square matrices.

1

3. See Page 8 for worked solutions.Write down the system of linear equations corresponding to the aug-

mented matrix A =

3 −8 11 0−1 0 8 12 −8 4 10

.

Find the elementary row operation that transfers A into B, and then findthe inverse elementary row operation that transfers B into A, where:

(a) B =

3 −8 11 0−1 0 8 11 −4 2 5

(b) B =

−1 0 8 13 −8 11 02 −8 4 10

(c) B =

1 −8 27 2−1 0 8 12 −8 4 10

4. See Page 10 for worked solutions.

Consider the following system of linear equations:

x2 − 5x3 = 6

2x1 − 2x2 + 11x3 = −12

4x1 − 3x2 + 17x3 = −18

(a) Use Gauss-Jordan elimination to put the augmented matrix corre-sponding to this system into reduced row echelon form.

(b) State which variables are free and which are basic.

(c) Find any solutions of the system.

(d) Give a geometric interpretation.

5. See Page 13 for worked solutions.Which of the following are linear combinations of u = (1, 1, 2)t and v =(1, 3,−1)t?

(a) (0, 0, 0)t (b) (1, 1, 1)t (c) (1,−3, 8)t

6. See Page 15 for worked solutions.

Find the general solution to Ax = b when A =

2 −4 −2 −36 −12 −2 −7−1 2 4 3

and also:

(a) b =

000

(b) b =

−1−12

(c) b =

−7−15

8

(d) b =

−227

Hint: For parts (b), (c) and (d), it may be helpful to find a particularsolution first!

2

7. See Page 18 for worked solutions.Let A be a 3 × 5 matrix.

(a) Write down the elementary matrices which, when multiplied on theleft of A, will perform the following elementary row operations:

(i) Subtract 2 times row 1 from row 2;

(ii) Interchange row 2 and row 3;

(iii) Multiply row 3 by −15.

(b) Find the matrix E which, when multiplied on the left of A, willperform all three row operations from part (a) in order.

8. See Page 20 for worked solutions.

Write A =

[3 41 2

]as a product of elementary matrices.

9. See Page 21 for worked solutions.

Let B =

1 2 1−2 −3 −14 11 −2

. Find B−1 using row operations.

3

1. Click here to go to question list.

Let A =

[0 1 −1−1

20 1

2

], B =

305

, C =[1 −1 1

]and D =

[10 05 2.5

].

Also let I be the 3 × 3 identity.For each of the following, state if the operation is defined and calculatethe result if it is defined.

(a) BA (b) AB (c) AtD (d) DAt

(e) BC (f) CB (g) B + C (h) (Bt + C)(BC + I)

Link to video on YouTube

4

5

6

2. Click here to go to question list.Show that if A, B are (any) two matrices such that AB and BA are bothdefined then AB and BA are both square matrices.

Link to video on YouTube

7

3. Click here to go to question list.Write down the system of linear equations corresponding to the aug-

mented matrix A =

3 −8 11 0−1 0 8 12 −8 4 10

.

Find the elementary row operation that transfers A into B, and then findthe inverse elementary row operation that transfers B into A, where:

(a) B =

3 −8 11 0−1 0 8 11 −4 2 5

(b) B =

−1 0 8 13 −8 11 02 −8 4 10

(c) B =

1 −8 27 2−1 0 8 12 −8 4 10

Link to video on YouTube

8

9

4. Click here to go to question list.Consider the following system of linear equations:

x2 − 5x3 = 6

2x1 − 2x2 + 11x3 = −12

4x1 − 3x2 + 17x3 = −18

(a) Use Gauss-Jordan elimination to put the augmented matrix corre-sponding to this system into reduced row echelon form.

(b) State which variables are free and which are basic.

(c) Find any solutions of the system.

(d) Give a geometric interpretation.

Link to video on YouTube

10

11

12

5. Click here to go to question list.Which of the following are linear combinations of u = (1, 1, 2)t and v =(1, 3,−1)t?

(a) (0, 0, 0)t (b) (1, 1, 1)t (c) (1,−3, 8)t

Link to video on YouTube

13

14

6. Click here to go to question list.

Find the general solution to Ax = b when A =

2 −4 −2 −36 −12 −2 −7−1 2 4 3

and also:

(a) b =

000

(b) b =

−1−12

(c) b =

−7−15

8

(d) b =

−227

Hint: For parts (b), (c) and (d), it may be helpful to find a particularsolution first!

Link to video on YouTube

15

16

17

7. Click here to go to question list.Let A be a 3 × 5 matrix.

(a) Write down the elementary matrices which, when multiplied on theleft of A, will perform the following elementary row operations:

(i) Subtract 2 times row 1 from row 2;

(ii) Interchange row 2 and row 3;

(iii) Multiply row 3 by −15.

(b) Find the matrix E which, when multiplied on the left of A, willperform all three row operations from part (a) in order.

Link to video on YouTube

18

19

8. Click here to go to question list.

Write A =

[3 41 2

]as a product of elementary matrices.

Link to video on YouTube

20

9. Click here to go to question list.

Let B =

1 2 1−2 −3 −14 11 −2

. Find B−1 using row operations.

Link to video on YouTube

21

22

10. Let A, B and C be n×n matrices such that AB = I and BC = I. Provethat A, B and C are all invertible.

Link to video on YouTube

23