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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⎡
⎣⎢
⎤
⎦⎥