Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

2 2 matrix:a b

Ac d

Using Matrices to Solve Simultaneous Equations

2 2 matrix:a b

Ac d

Determinant of A: det A A ad bc

Using Matrices to Solve Simultaneous Equations

2 2 matrix:a b

Ac d

Determinant of A: det A A ad bc

Adjoint matrix: adjd b

Ac a

Using Matrices to Solve Simultaneous Equations

2 2 matrix:a b

Ac d

Determinant of A: det A A ad bc

Adjoint matrix: adjd b

Ac a

a bc d

swap

Using Matrices to Solve Simultaneous Equations

2 2 matrix:a b

Ac d

Determinant of A: det A A ad bc

Adjoint matrix: adjd b

Ac a

a bc d

swap

change signs

Using Matrices to Solve Simultaneous Equations

2 2 matrix:a b

Ac d

Determinant of A: det A A ad bc

Inverse of A:

Adjoint matrix: adjd b

Ac a

a bc d

swap

change signs

1 adjAAA

Using Matrices to Solve Simultaneous Equations

2 2 matrix:a b

Ac d

Determinant of A: det A A ad bc

Inverse of A:x m

Ay n

1x mA

y n

Adjoint matrix: adjd b

Ac a

a bc d

swap

change signs

1 adjAAA

Using Matrices to Solve Simultaneous Equations

2 2 matrix:a b

Ac d

Determinant of A: det A A ad bc

Inverse of A:x m

Ay n

1x mA

y n

Adjoint matrix: adjd b

Ac a

a bc d

swap

change signs

1 adjAAA

If 0, then lines are paralleli.e. no solutions

A

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

2 3 215 2 3

xy

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

2 3 215 2 3

xy

2 2 5 3A 2 53 2

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

2 3 215 2 3

xy

2 2 5 3A 2 53 2

2adj

2A

1. swap

2 3adj

5 2A

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

2 3 215 2 3

xy

2 2 5 3A 2 53 2

1. swap

2.change signs

2 3adj

5 2A

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

2 3 215 2 3

xy

2 2 5 3A

2 3 21111 5 2 3

xy

2 53 2

1. swap

2.change signs

2 3adj

5 2A

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

2 3 215 2 3

xy

2 2 5 3A

2 3 21111 5 2 3

xy

Multiply the row of the matrix with the column

of the vector

2 21 3 3111 5 21 2 3

xy

33111 99

xy

2 53 2

1. swap

2.change signs

2 3adj

5 2A

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

2 3 215 2 3

xy

2 2 5 3A

2 3 21111 5 2 3

xy

Multiply the row of the matrix with the column

of the vector

2 21 3 3111 5 21 2 3

xy

33111 99

xy

39

xy

2 53 2

1. swap

2.change signs

2 3adj

5 2A

e.g. ( ) 2 3 21 (1)5 2 3 (2)

i x yx y

2 3 215 2 3

xy

2 2 5 3A

2 3 21111 5 2 3

xy

Multiply the row of the matrix with the column

of the vector

3, 9x y

2 21 3 3111 5 21 2 3

xy

33111 99

xy

39

xy

2 53 2

1. swap

2.change signs

( ) 4 5 3 (1)3 7 13 (2)

ii x yx y

( ) 4 5 3 (1)3 7 13 (2)

ii x yx y

4 5 33 7 13

xy

7 5 31

43 3 4 13xy

( ) 4 5 3 (1)3 7 13 (2)

ii x yx y

4 5 33 7 13

xy

7 5 31

43 3 4 13xy

7 3 5 13143 3 3 4 13

xy

86143 43

xy

( ) 4 5 3 (1)3 7 13 (2)

ii x yx y

4 5 33 7 13

xy

7 5 31

43 3 4 13xy

2, 1x y

7 3 5 13143 3 3 4 13

xy

21

xy

86143 43

xy

3 3 matrix:a b c

A d e fg h i

3 3 matrix:a b c

A d e fg h i

Determinant of A: det A A aei bfg cdh gec hfa idb

3 3 matrix:a b c

A d e fg h i

Determinant of A: det A A aei bfg cdh gec hfa idb

Adjoint matrix: adj

te f d f d eh i g i g hb c a c a b

Ah i g i g h

b c a c a be f d f d e

3 3 matrix:a b c

A d e fg h i

Determinant of A: det A A aei bfg cdh gec hfa idb

Adjoint matrix: adj

te f d f d eh i g i g hb c a c a b

Ah i g i g h

b c a c a be f d f d e

a b cd e fg h i

1. cover up row and column of the position looking for

3 3 matrix:a b c

A d e fg h i

Determinant of A: det A A aei bfg cdh gec hfa idb

Adjoint matrix: adj

te f d f d eh i g i g hb c a c a b

Ah i g i g h

b c a c a be f d f d e

a b cd e fg h i

1. cover up row and column of the position looking for

2. find determinant of the remaining matrix

3 3 matrix:a b c

A d e fg h i

Determinant of A: det A A aei bfg cdh gec hfa idb

Adjoint matrix: adj

te f d f d eh i g i g hb c a c a b

Ah i g i g h

b c a c a be f d f d e

t = transpose rows and columns

a b cd e fg h i

1. cover up row and column of the position looking for

2. find determinant of the remaining matrix

e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)

x y zx y zx y z

e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)

x y zx y zx y z

1 2 1 52 3 4 284 5 3 10

xyz

e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)

x y zx y zx y z

1 2 1 52 3 4 284 5 3 10

xyz

1 2 12 3 44 5 3

A

e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)

x y zx y zx y z

1 2 1 52 3 4 284 5 3 10

xyz

1 2 12 3 44 5 3

A

1 2 1 1 22 3 4 2 34 5 3 4 5

e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)

x y zx y zx y z

1 2 1 52 3 4 284 5 3 10

xyz

1 2 12 3 44 5 3

A

1 2 1 1 22 3 4 2 34 5 3 4 5

1 3 3 2 4 4 1 2 5

9 32 10

e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)

x y zx y zx y z

1 2 1 52 3 4 284 5 3 10

xyz

1 2 12 3 44 5 3

A

1 2 1 1 22 3 4 2 34 5 3 4 5

1 3 3 2 4 4 1 2 5

9 32 10 4 3 1 5 4 1 3 2 2

12 20 12 11

3 4 2 4 2 35 3 4 3 4 52 1 1 1 1 2

adj5 3 4 3 4 52 1 1 1 1 23 4 2 4 2 3

t

A

1 2 12 3 44 5 3

A

3 4 2 4 2 35 3 4 3 4 52 1 1 1 1 2

adj5 3 4 3 4 52 1 1 1 1 23 4 2 4 2 3

t

A

1 2 12 3 44 5 3

A

11 22 221 1 35 6 7

t

3 4 2 4 2 35 3 4 3 4 52 1 1 1 1 2

adj5 3 4 3 4 52 1 1 1 1 23 4 2 4 2 3

t

A

1 2 12 3 44 5 3

A

11 22 221 1 35 6 7

t

11 1 522 1 622 3 7

1 2 1 52 3 4 284 5 3 10

xyz

1 2 1 52 3 4 284 5 3 10

xyz

11 1 5 51 22 1 6 28

1122 3 7 10

xyz

1 2 1 52 3 4 284 5 3 10

xyz

11 1 5 51 22 1 6 28

1122 3 7 10

xyz

11 5 1 28 5 101 22 5 1 28 6 10

1122 5 3 28 7 10

xyz

331 22

1144

xyz

1 2 1 52 3 4 284 5 3 10

xyz

11 1 5 51 22 1 6 28

1122 3 7 10

xyz

11 5 1 28 5 101 22 5 1 28 6 10

1122 5 3 28 7 10

xyz

331 22

1144

xyz

32

4

xyz

3, 2, 4x y z

1 2 1 52 3 4 284 5 3 10

xyz

11 1 5 51 22 1 6 28

1122 3 7 10

xyz

11 5 1 28 5 101 22 5 1 28 6 10

1122 5 3 28 7 10

xyz

331 22

1144

xyz

32

4

xyz

3, 2, 4x y z

Worksheet+

try some ofExercise 1H;

1, 2, 6