Matrices

Question 1

The order of a matrix is given as …

Answer 1

n x m

Number of rows x Number of columns

Question 2

What is a square matrix?

Answer 2

A square matrix is a matrix with the same number of rows as columns. i.e. 2x2, 3x3, etc.

Question 3

How does one add/subtract matrices?

Answer 3

1. To add or subtract matrices they must be of the same order. i.e. 2x2 add/subtract 2x2.
2. Simply add or subtract corresponding elements.

Question 4

How does one multiply a matrix by a scalar?

Answer 4

To multiply a matrix by a scalar simply multiply each element of the matrix by the scalar/number.

Question 5

How does one multiply matrices together?

Answer 5

1. In order to multiply two matrices together the order must correspond as follows:A X B = C
   (nxm) (mxk) (nxk)

The number of columns of the first matrix and number of rows of the second matrix (based on order of multiplication) must match for the matrices to be multiplied together.

The order of the resulting matrix can be determined from the outside values of the original matrices.

Question 6

Matrix multiplication is …

Answer 6

Not commutative. i.e AB does not equal BA.

Unless the question tells you otherwise

Question 7

What is the identity matrix?

Answer 7

1. It’s always square, i.e. 2x2 or 3x3 etc.
2. (1. 0.) (1. 0. 0.)
   (0. 1.) (0. 1. 0.) etc.
   (0. 0. 1.)

Question 8

What effect does the identity matrix have on a matrix when they are multiplied together?

Answer 8

No effect.

IxA = AxI = A

Question 9

What is the zero matrix?

Answer 9

Also known as the null matrix.

Effectively a matrix where all elements are zero, regardless of whether it is a square matrix or not.

Question 10

Name four transformations.

Answer 10

Rotation
Reflection
Enlargement
Translation

Question 11

What is a transformation?

Answer 11

A transformation moves all the points (x,y) in a plane, according to some rule.

Question 12

What is the name of the new point to which (x) has moved?

(y)

Answer 12

The image of (x), i.e. (x’)

(y) (y’)

Question 13

What is a linear transformation?

Answer 13

A linear transformation has the special properties:

1. The transformation only involves linear expressions of x and y. i.e kx or ky or kx +ky. It does not contain any x2 or y2 or xy.
2. The origin (0,0) is not moved by the transformation.

Question 14

Linear transformations can be expressed as a pair of simultaneous equations in the form:

Answer 14

x' = ax + cy
y' = bx + dy

Question 15

How can you determine whether a transformation is linear or not?

Answer 15

Two steps:

1. Look to see if it contains any x2 or y2 or xy values.
2. Judge its effect on the origin (0,0), the origin should not move.

Question 16

How can we identify the matrix representing a particular transformation?

Answer 16

In order to identify the matrix representing a particular transformation you should consider the effect of the matrix/transformation on the points with position vectors į (1)and j (0)
(0) (1)

Question 17

What is the matrix which represents an anti-clockwise rotation through any angle Ø? Can this matrix be applied to a clockwise rotation?

Answer 17

(CosØ, -SinØ)
(SinØ, CosØ)

Yes, if the clockwise angle is subtracted from 360° to get an anti-clockwise angle.

Question 18

How can this equation be used to find the angle of rotation (and direction) when only the transformation matrix is known?

Answer 18

Match the elements up with the corresponding elements of the equation. Solve for Ø, making sure to remove any negatives if present.
After finding Ø, look at the original sign of the elements in the first column of the transformation matrix, this will tell you whether sin or cosin are positive and/or negative in relation to the CAST diagram.
The angle Ø is always drawn from the X-axis.

Question 19

When multiplying points by a transformation matrix, the matrix is always put …

Answer 19

First, i.e. before the points.

Question 20

The transformation matrix X is applied to the points P, then the transformation matrix Q is applied to the new points. Represent this information algebraically.

Answer 20

QXP

Question 21

What is an invariant point?

Answer 21

An invariant point for a transformation is a point which is mapped to itself by the transformation.

Question 22

For a linear transformation, name an invariant point.

Answer 22

The origin (0,0)

Question 23

If (x,y) is an invariant point of the matrix (a, c), form two
(b, d)
simultaneous equations.

Answer 23

ax + cy = x

bx + dy = y

Question 24

How does one find the invariant points for a transformation?

Answer 24

Find out whether the two simultaneous equations are equal to each other, by rearranging to make x or y the subject. If they are equal, then they represent a line of invariant points. If they are not, the only invariant point is the origin (0,0).
The equation of the line of invariant points is the equation formed in terms of x after rearrangement (when the two equations are equal).

Question 25

What is an invariant line?

Answer 25

An invariant line for a transformation is a line which is mapped to itself by the transformation.

Question 26

What is the inverse of a rotation?

Answer 26

The inverse of a rotation is a rotation through the same angle about the same point but in the opposite direction.

Question 27

What is the inverse of a reflection?

Answer 27

The inverse of a reflection is another reflection in the same line.

Question 28

What is the inverse of an enlargement?

Answer 28

The inverse of an enlargement with scale factor K is an enlargement with scale factor 1/K.

Question 29

How can one map a point back to its original position after the effects of a transformation matrix.

Answer 29

Multiply the new points by the inverse of the original transformation matrix to map the points back to their original position.