VCE Maths Methods - Unit 2 - Matrices

Matrices

• Introduction to matrices • Addition & subtraction• Scalar multiplication• Matrix multiplication• The unit matrix• Matrix division - the inverse matrix• Using matrices - simultaneous equations• Matrix transformations

1

VCE Maths Methods - Unit 2 - Matrices

Matrices

• A matrix is an array of individual elements.

• The order (dimensions) of a matrix is de!ned by the number of rows & columns.

2

1 2−4 6

⎡

⎣⎢

⎤

⎦⎥

2459

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

2 x 2 matrix

4 x 1 matrix

3 0 −3⎡⎣ ⎤⎦ 1 x 3 matrix

rows × columns = order

VCE Maths Methods - Unit 2 - Matrices

Examples of matrices

• The daily rate for hiring cars:

3

1 day 2 - 7 days 8 + days

Kia Rio $120 $105 $90

Toyota Camry $140 $125 $110

Holden Statesman $170 $145 $120

R =120 105 90140 125 110170 145 120

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

VCE Maths Methods - Unit 2 - Matrices

Addition & subtraction of matrices

4

• A + B = C

• A & B must be of the same order.

• Corresponding elements in A & B are added or subtracted.

• C has the same order as A & B.

• The commutative law holds for matrices: A + B = B + A

• eg a $10 holiday surcharge applied to the car rental:

R =120 105 90140 125 110170 145 120

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

S =10 10 1010 10 1010 10 10

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

R+S =130 115 100150 135 120180 155 130

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

VCE Maths Methods - Unit 2 - Matrices

Scalar multiplication

5

• All elements can be multiplied by a scalar (single number).

• eg a 20% increase in the cost of hire cars:

Rnew =1.2×Rold

Rold =120 105 90140 125 110170 145 120

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Rnew =144 126 108168 150 132214 174 144

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

VCE Maths Methods - Unit 2 - Matrices

Matrix multiplication

6

• A x B = C

• Rows in the !rst matrix multiply by the columns in the second.

• The number of rows in A & the number of columns in B gives the dimensions of C .

• The number of columns in A must match the number of rows in B.

• (m x n) (n x p) gives an (m x p) matrix.

• In general, B x A ≠ C.

2 4⎡⎣ ⎤⎦ 3

5⎡

⎣⎢

⎤

⎦⎥= (2×3)+(4×5)[ ]= 26[ ]

VCE Maths Methods - Unit 2 - Matrices

=(3×4)+(0×1

2) (3×−1)+(0×−3)

(1×4)+(−2×12

) (1×−1)+(−2×−3)

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Matrix multiplication

7

3 01 −2

⎡

⎣⎢

⎤

⎦⎥×

4 −112

−3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

= 12+0 −3+0

4−1 −1+6⎡

⎣⎢

⎤

⎦⎥=

12 −33 5

⎡

⎣⎢

⎤

⎦⎥

=

(a11×b11)+(a12×b21) (a11×b12 )+(a12×b22 )(a21×b11)+(a22×b21) (a21×b12 )+(a22×b22 )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

a11 a12

a21 a22

⎡

⎣⎢⎢

⎤

⎦⎥⎥×

b11 b12

b21 b22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

• Rows multiply by columns: The number of rows in A & the number of columns in B gives the dimension of C .

• The number of columns in A must match the number of rows in B.

VCE Maths Methods - Unit 2 - Matrices

Possible matrix multiplications

8

• Rows multiply by columns: The number of rows in A & the number of columns in B gives the dimension of C .

• The number of columns in A must match the number of rows in B.

3 2 1⎡⎣ ⎤⎦ 024

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1 23 4

⎡

⎣⎢

⎤

⎦⎥ −2 5

0 7⎡

⎣⎢

⎤

⎦⎥

1 x 3 3 x 1 1 x 12 x 2 2 x 2 2 x 2

024

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3 2 1⎡⎣ ⎤⎦

3 x 1 1 x 3 3 x 3

_ _ _⎡⎣

⎤⎦

___

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= _[ ]

10 9 87 6 5

⎡

⎣⎢

⎤

⎦⎥

123

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

2 x 3 3 x 1 2 x 1

=

_ __ _

⎡

⎣⎢⎢

⎤

⎦⎥⎥ = _[ ]

=_ _ __ _ __ _ _

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=__

⎡

⎣⎢⎢

⎤

⎦⎥⎥

-2 19

-6 438

0 0 06 4 2

12 8 4

52

34

VCE Maths Methods - Unit 2 - Matrices

The unit matrix

9

• The unit matrix (I) is a square matrix that can be multiplied by another matrix (A) to not alter that matrix.

• AI = IA = A if A is a square matrix.

• Non square matrices can be multiplied by a square identity matrix.

=

(2×1)+(3×0) (2×0)+(3×1)(6×1)+(2×0) (6×0)+(2×1)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1 00 1

⎡

⎣⎢

⎤

⎦⎥ 2 3

6 2⎡

⎣⎢

⎤

⎦⎥

2 36 2

⎡

⎣⎢

⎤

⎦⎥ 1 0

0 1⎡

⎣⎢

⎤

⎦⎥

= 2 3

6 2⎡

⎣⎢

⎤

⎦⎥

= 2 3

6 2⎡

⎣⎢

⎤

⎦⎥

4 5 0−3 6 6

⎡

⎣⎢

⎤

⎦⎥

1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ = 4 5 0

−3 6 6⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Unit 2 - Matrices

Matrix division - the inverse matrix

10

• A square matrix has an inverse matrix A-1 , where A x A-1 = I.

• Multiplying by A-1 is equivalent to division.

• For a 2 x 2 matrix:

a bc d

⎡

⎣⎢

⎤

⎦⎥

−1

= 1

ad −bc d −b

−c a⎡

⎣⎢

⎤

⎦⎥

2 x 2 Matrix determinant (det A) = ad - bc

• If det A = 0, no solution exists.

• If both rows of the matrix are multiples of each other, then the determinant will be zero. (A singular matrix)

VCE Maths Methods - Unit 2 - Matrices

Matrix division - the inverse matrix

11

• For example, the matrices shown below:

2 3−3 5

⎡

⎣⎢

⎤

⎦⎥

−1

Det A = 0, no solution exists. = 1

19 5 −3

3 2⎡

⎣⎢

⎤

⎦⎥

=

519

−319

319

219

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

2 36 9

⎡

⎣⎢

⎤

⎦⎥

−1

= 1

10−−9 5 −3

3 2⎡

⎣⎢

⎤

⎦⎥

= 1

18−18 9 −3−6 2

⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Unit 2 - Matrices

Using matrices - simultaneous equations

12

• Matrices can be used to help solve simultaneous equations of two or more variables.

• For example, !nding the equation of a quadratic curve (y = ax2 + bx +c) that passes through three points (-1,6) , (0, 3) & (2, 9).

1 −1 10 0 14 2 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

abc

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=639

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

6= a(−1)2+b(−1)+c

3= a(0)2+b(0)+c

9= a(2)2+b(2)+c

1 −1 10 0 14 2 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−1639

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

abc

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

16

2 −3 1−4 3 10 6 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

639

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

abc

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

2−13

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

= abc

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

y =2x2 −x+3

a b c

VCE Maths Methods - Unit 2 - Matrices

Matrix transformations - translations

13

• Matrix operations can be used to !nd the transformations of points.

• These can be translations, re#ections, rotations or dilations.

y intercept: x = 0

y

x

(1,2)(5,3)

Translations: The point can be moved across or up / down.

x’ = x + a

+4+1

y’ = y + b

xy

⎡

⎣⎢

⎤

⎦⎥+

ab

⎡

⎣⎢

⎤

⎦⎥=

x 'y '

⎡

⎣⎢

⎤

⎦⎥

12

⎡

⎣⎢

⎤

⎦⎥+

41

⎡

⎣⎢

⎤

⎦⎥=

53

⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Unit 2 - Matrices

Matrix transformations - re!ections

14

y intercept: x = 0

y

x

(1,2)

(2,1)

Re#ection around the y = x line: the x & y co-ordinates are swapped.

x’ = yy’ = x

0 11 0

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢

⎤

⎦⎥=

x 'y '

⎡

⎣⎢

⎤

⎦⎥

0 11 0

⎡

⎣⎢

⎤

⎦⎥ 1

2⎡

⎣⎢

⎤

⎦⎥

=

(1×0)+(1×2)(1×1)+(0×2)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 2

1⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Unit 2 - Matrices

Matrix transformations - re!ections

15

y intercept: x = 0

y

x

(1,2)

(1,-2)

Re#ection around the x axis: y value changes sign.

x’ = xy’ = -y

1 00 −1

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢

⎤

⎦⎥=

x 'y '

⎡

⎣⎢

⎤

⎦⎥

1 00 −1

⎡

⎣⎢

⎤

⎦⎥ 1

2⎡

⎣⎢

⎤

⎦⎥

=

(1×1)+(0×2)(0×1)−(1×2)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 1

−2⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Unit 2 - Matrices

Matrix transformations - re!ections

16

y intercept: x = 0

y

x

(1,2)(-1,2)

Re#ection around the y axis: x value changes sign.

x’ = -xy’ = y

−1 00 1

⎡

⎣⎢

⎤

⎦⎥ x

y⎡

⎣⎢

⎤

⎦⎥=

x 'y '

⎡

⎣⎢

⎤

⎦⎥

−1 00 1

⎡

⎣⎢

⎤

⎦⎥ 1

2⎡

⎣⎢

⎤

⎦⎥

=

(−1×1)+(0×2)(0×1)+(1×2)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= −1

2⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Unit 2 - Matrices

Matrix transformations - dilations

17

y intercept: x = 0

y

x

(1,2) (5,2)

Dilation from the y axis: x value is multiplied.

x’ = 5x = 5y’ = y

k 00 1

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢

⎤

⎦⎥=

x 'y '

⎡

⎣⎢

⎤

⎦⎥

5 00 1

⎡

⎣⎢

⎤

⎦⎥ 1

2⎡

⎣⎢

⎤

⎦⎥

=

(5×1)+(0×2)(0×1)+(1×2)

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

52

⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Unit 2 - Matrices

Matrix transformations - dilations

18

y intercept: x = 0

y

x

(1,2)

(1,4)

Dilation from the x axis: y value is multiplied.

x’ = xy’ = 2y = 4

1 00 k

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢

⎤

⎦⎥=

x 'y '

⎡

⎣⎢

⎤

⎦⎥

1 00 2

⎡

⎣⎢

⎤

⎦⎥ 1

2⎡

⎣⎢

⎤

⎦⎥

=

(1×1)+(0×2)(0×1)+(2×2)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 1

4⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Unit 2 - Matrices

Matrix transformations - rotations

19

y intercept: x = 0

y

x

(1,2)(-2,1)

Anti-clockwise rotation about the origin.

x’ = cos90°x - sin90°yy’ = sin90°x + cos90°y

cosθ −sinθsinθ cosθ

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢

⎤

⎦⎥=

x 'y '

⎡

⎣⎢

⎤

⎦⎥

cos90° −sin90°sin90° cos90°

⎡

⎣⎢

⎤

⎦⎥ 1

2⎡

⎣⎢

⎤

⎦⎥

=

(0×1)−(1×2)(1×1)+(0×2)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= −2

1⎡

⎣⎢

⎤

⎦⎥