Chapter 6 Matrices Flashcards
What does every matrix have?
Numbers (elements)
Collumn/s
Row/s
How do you workout out the order of a matrix?
The order of a matrix is the number of rows (m) x the number of columns (n).
So a matrix with 2 rows and 1 column has the order ‘2 x 1’
When can you add and subtract matrices?
Only when they have the same order.
How do you add and subtract matrices?
By adding the corresponding elements from each matrix.
E.g. (1 2) + (4 5) = (5 7)
What do you do to multiply a matrix by any number / when you see a big number in front of a matrix?
Multiply each element in the matrix by that number.
E.g. 4 (3 2) = (12 8)
What is the scalar quantity?
The number you multiply your matrix by.
What is the matrix quantity?
The product of a whole multiplication (a number x a matrix).
How do you work out the values of letters (missing numbers) in matrix additions / subtractions?
Make an equation for the letter.
E.g. (a 7) + (3 -1) = (11 6)
a + 3 = 11
a = 11 - 3
a = 8
When can you multiply matrices?
Only when the number of columns in the first matrix is equal to the number of rows in the second matrix.
Number of collumns = second number in order
Number of rows = first number in order
Or just write out the order of each matrix and if the two inside numbers are the same then can multiply!
E.g. 1 x 2 2 x 2
Inside numbers are the same so can multiply!
How can you workout the order of the product of a matrix multiplication?
Write the the order of both the matrices you are multiplying,
E.g. 2 x 2 2 x 1
The order of your product will be the two outside numbers!
So 2 x 1
How do you multiply matrices?
1) workout the order of your product
2) first no. In product = (first number in first matrix x first number in second) + (second number in first matrix x third number in second matrix)
3) second no. In product = (first number in first matrix x second number in second) + (second number in first matrix x fourth number in second matrix)
4) third no. In product = (third number in first matrix x first number in second matrix) + (fourth number in first matrix x third number in second matrix)
5) fourth no. In product = (third number in first matrix x second number in second matrix) + (fourth number in first matrix x fourth number in second matrix)
and etc.
Some matrices may have less or more than four numbers.
What is the identity matrix?
Is always called ‘I’.
= 1 0
0 1
Represents the number one as when multiplied by a matrix it stays the same, but when added to a number it adds one.
What is the zero matrix?
Is always called ‘Z’.
= 0 0
0 0
Represents the number 0, as when multiplied by a matrix it changes it to zero, when added to a matrix it stays the same.
What is a matrix?
A matrix is a rectangular array of numbers.
How can you work out the value of letters (missing numbers) in matrix multiplications?
- work out what was multiplied by what to make each element in the product matrix using the matrix multiplication method.
- form equations to solve each letter
- solve them.
How do you work out the image of a point after transformation by a matrix?
E.g. Point A = (1,1)
Matrix = 2 0
0 2
- turn A into a vector = 1
1 - multiply by Matrix (matrix always in front of vector in multiplication)
- so image of point A after transformation by the matrix is the product matrix of the multiplication.
How do you transform the unit square (coordinates (0,0) (1,0) (1,1) (0,1) ) using a matrix?
- Multiply all 4 vertices (coordinates) of the unit square by your matrix.
- Plot each new point to make new square.
- work out the total enlargement from the original unit square to your new square.
What matrix rotates something 90 degrees clockwise about 0?
0 1
-1 0
What matrix rotates something 180 degrees about 0?
-1 0
0 -1
What matrix rotates something 90 degrees anti-clockwise about 0?
0 -1
1 0
What matrix reflects something in the x axis?
1 0
0 -1
what matrix reflects something in the y axis?
-1 0
0 1
What matrix reflects something in the line y=x?
0 1
1 0
What matrix reflects something in the line y=-x?
0 -1
-1 0
What matrix enlarges something by scale factor m, centre 0?
m 0
0 m
If asked to describe fully the single transformation a matrix represents…
Work out the image of the two points A (1,0) and B (0,1) under transformation by that matrix every time
Draw the old A and B points on one graph, and the new A’ and B’ points on a graph next to it to work out what had happened to them e.g. Reflection / rotation / enlargement etc
