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
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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.