Chapter 6 Matrices

Question 1

What does every matrix have?

Answer 1

Numbers (elements)
Collumn/s
Row/s

Question 2

How do you workout out the order of a matrix?

Answer 2

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’

Question 3

When can you add and subtract matrices?

Answer 3

Only when they have the same order.

Question 4

How do you add and subtract matrices?

Answer 4

By adding the corresponding elements from each matrix.

E.g. (1 2) + (4 5) = (5 7)

Question 5

What do you do to multiply a matrix by any number / when you see a big number in front of a matrix?

Answer 5

Multiply each element in the matrix by that number.

E.g. 4 (3 2) = (12 8)

Question 6

What is the scalar quantity?

Answer 6

The number you multiply your matrix by.

Question 7

What is the matrix quantity?

Answer 7

The product of a whole multiplication (a number x a matrix).

Question 8

How do you work out the values of letters (missing numbers) in matrix additions / subtractions?

Answer 8

Make an equation for the letter.

E.g. (a 7) + (3 -1) = (11 6)

a + 3 = 11
a = 11 - 3
a = 8

Question 9

When can you multiply matrices?

Answer 9

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!

Question 10

How can you workout the order of the product of a matrix multiplication?

Answer 10

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

Question 11

How do you multiply matrices?

Answer 11

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.

Question 12

What is the identity matrix?

Answer 12

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.

Question 13

What is the zero matrix?

Answer 13

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.

Question 14

What is a matrix?

Answer 14

A matrix is a rectangular array of numbers.

Question 15

How can you work out the value of letters (missing numbers) in matrix multiplications?

Answer 15

• 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.

Question 16

How do you work out the image of a point after transformation by a matrix?

Answer 16

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.

Question 17

How do you transform the unit square (coordinates (0,0) (1,0) (1,1) (0,1) ) using a matrix?

Answer 17

• 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.

Question 18

What matrix rotates something 90 degrees clockwise about 0?

Answer 18

0 1

-1 0

Question 19

What matrix rotates something 180 degrees about 0?

Answer 19

-1 0

0 -1

Question 20

What matrix rotates something 90 degrees anti-clockwise about 0?

Answer 20

0 -1

1 0

Question 21

What matrix reflects something in the x axis?

Answer 21

1 0

0 -1

Question 22

what matrix reflects something in the y axis?

Answer 22

-1 0

0 1

Question 23

What matrix reflects something in the line y=x?

Answer 23

0 1

1 0

Question 24

What matrix reflects something in the line y=-x?

Answer 24

0 -1

-1 0

Question 25

What matrix enlarges something by scale factor m, centre 0?

Answer 25

m 0

0 m

Question 26

If asked to describe fully the single transformation a matrix represents…

Answer 26

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