FMAT pure1 matrices and transformations

Question 1

What is a matrix?

Answer 1

A matrix is a rectangular array of numbers arranged in rows and columns. It is used to store information.

Question 2

How are matrices classified?

Answer 2

Matrices are classified by their dimensions (order), written as m × n:
- m: Number of rows,
- n: Number of columns.

Question 3

What is a square matrix?

Answer 3

A matrix with the same number of rows and columns (e.g., 3 × 3).

Question 4

What are the conditions for adding or subtracting matrices?

Answer 4

Matrices must have the same dimensions (order) to be added or subtracted.

Question 5

Example: Add and subtract matrices A and B where:
A = [[2, 3], [-1, 0]], B = [[3, 4], [-1, -2]].

Answer 5

Addition: A + B = [[2 + 3, 3 + 4], [-1 + (-1), 0 + (-2)]] = [[5, 7], [-2, -2]]
Subtraction: A - B = [[2 - 3, 3 - 4], [-1 - (-1), 0 - (-2)]] = [[-1, -1], [0, 2]]

Question 6

How is scalar multiplication of a matrix performed?

Answer 6

Each element of the matrix is multiplied by the scalar.
Example: For A = [[1, 2], [3, 4]] and scalar 3:
3A = [[3 * 1, 3 * 2], [3 * 3, 3 * 4]] = [[3, 6], [9, 12]]

Question 7

What is the condition for multiplying two matrices?

Answer 7

Matrix multiplication is possible when the number of columns in the first matrix equals the number of rows in the second matrix.

Question 8

Example: Multiply matrices A and B where:
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]

Answer 8

The product AB is:
AB = [[15 + 27, 16 + 28], [35 + 47, 36 + 48]]
= [[19, 22], [43, 50]]

Question 9

What are the key properties of matrix multiplication?

Answer 9

1. Matrix multiplication is not commutative: AB ≠ BA.
2. Matrix multiplication is associative: (AB)C = A(BC).
3. Matrix multiplication is distributive over addition: A(B + C) = AB + AC.

Question 10

What is the identity matrix, and how does it affect multiplication?

Answer 10

The identity matrix I is a square matrix with 1s on the diagonal and 0s elsewhere:
I = [[1, 0], [0, 1]].
For any matrix A, IA = AI = A.

Question 11

Example: Find the product of A and the identity matrix where:
A = [[2, 3], [4, 5]], I = [[1, 0], [0, 1]].

Answer 11

AI = [[21 + 30, 20 + 31], [41 + 50, 40 + 51]]
= [[2, 3], [4, 5]]
AI = A

Question 12

What is a linear transformation in two dimensions?

Answer 12

A linear transformation maps a point (x, y) to (x’, y’) using:
x’ = ax + by
y’ = cx + dy
This can be written as a matrix equation:
[[x’], [y’]] = [[a, b], [c, d]] × [[x], [y]]

Question 13

How can you find the image of a shape under a transformation matrix?

Answer 13

Write the points of the shape as a matrix, multiply by the transformation matrix, and solve for the resulting points.

Question 14

Example: Find the image of A(3, 1), B(-2, 4), C(5, -1) under the transformation matrix [[2, 3], [-1, 0]].

Answer 14

1. Points matrix: [[3, -2, 5], [1, 4, -1]]
2. Multiply by the transformation matrix:
   [[2, 3], [-1, 0]] × [[3, -2, 5], [1, 4, -1]]
3. Result:
   [[9, 8, 7], [-3, 2, -5]]
4. Image:
   A → (9, -3), B → (8, 2), C → (7, -5)

Question 15

What is the transformation matrix for a stretch parallel to the x-axis?

Answer 15

[[k, 0], [0, 1]]
scale factor k

Question 16

What is the transformation matrix for a stretch parallel to the y-axis?

Answer 16

[[1, 0], [0, k]]
scale factor k

Question 17

What is the transformation matrix for an enlargement centred at the origin?

Answer 17

[[k, 0], [0, k]]
scale factor k

Question 18

What is the matrix for a 90° anticlockwise rotation about the origin?

Answer 18

[[0, -1], [1, 0]]

Question 19

What is the matrix for a 90° clockwise rotation about the origin?

Answer 19

[[0, 1], [-1, 0]]

Question 20

What is the general matrix for an anticlockwise rotation through an angle θ?

Answer 20

[[cos(θ), -sin(θ)], [sin(θ), cos(θ)]]

Question 21

Example: Find the matrix for an anticlockwise rotation through 140° about the origin.

Answer 21

1. Use the general rotation matrix:
   [[cos(140°), -sin(140°)], [sin(140°), cos(140°)]]
2. Approximate values:
   [[-0.766, -0.642], [0.642, -0.766]]

Question 22

What is the matrix for a reflection in the x-axis?

Answer 22

[[1, 0], [0, -1]]

Question 23

What is the matrix for a reflection in the y-axis?

Answer 23

[[-1, 0], [0, 1]]

Question 24

What is the matrix for a reflection in the line y = x?

Answer 24

[[0, 1], [1, 0]]

Question 25

What is the matrix for a reflection in the line y = -x?

Answer 25

[[0, -1], [-1, 0]]

Question 26

What is the matrix for a shear parallel to the x-axis with factor k?

Answer 26

[[1, k], [0, 1]]

Question 27

What is the matrix for a shear parallel to the y-axis with factor k?

Answer 27

[[1, 0], [k, 1]]

Question 28

How are combined transformations represented using matrices?

Answer 28

The combined transformation is represented by multiplying the matrices of the individual transformations. Order matters: Apply the rightmost transformation first.

Question 29

Example: Find the matrix for a reflection in the x-axis followed by a 90° anticlockwise rotation.

Answer 29

1. Reflection in x-axis: [[1, 0], [0, -1]].
2. 90° rotation: [[0, -1], [1, 0]].
3. Combined: [[0, -1], [1, 0]] × [[1, 0], [0, -1]] = [[0, 1], [1, 0]].
   This represents a reflection in the line y = x.

Question 30

What is an invariant point in a transformation?

Answer 30

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

Question 31

What is the condition for finding invariant points in a 2x2 transformation matrix?

Answer 31

Solve the matrix equation:
[[a, c], [b, d]] * [[x], [y]] = [[x], [y]].
This results in two equations:
1. ax + cy = x,
2. bx + dy = y.

Question 32

What is the significance of invariant points in transformations?

Answer 32

For linear transformations:
- The origin is either the only invariant point.
- Or all invariant points lie on a straight line through the origin.
Finding invariant points can also help determine properties like the mirror line in a reflection.

Question 33

Example: Find the line of invariant points for the matrix [[2, 1], [2, 3]].

Answer 33

1. Solve:
   [[2, 1], [2, 3]] * [[x], [y]] = [[x], [y]].
   This expands to:
   2x + y = x and 2x + 3y = y.
2. Simplify:
   x + y = 0 (both equations are equivalent).
3. Solution:
   The line of invariant points is y = -x.

Question 34

What is an invariant line in a transformation?

Answer 34

An invariant line is a line that maps to itself under the transformation, but the points on the line are not necessarily invariant points.

Question 35

How do you find invariant lines for a 2x2 transformation matrix?

Answer 35

1. Assume the invariant line is y = mx + c.
2. Substitute a general point (x, mx + c) into the transformation matrix.
3. Solve for m and c by ensuring the image satisfies y = mx + c.

Question 36

What is the relationship between invariant points and invariant lines?

Answer 36

• A line of invariant points is also an invariant line.
• An invariant line is not necessarily a line of invariant points.

Question 37

Example: Find the invariant lines for the matrix [[2, 1], [2, 3]].

Answer 37

1. Assume y = mx + c.
2. Apply the transformation:
   [[2, 1], [2, 3]] * [[x], [mx + c]].
   This expands to:
   [[2x + mx + c], [2x + 3mx + 3c]].
3. Ensure the image satisfies y = mx + c:
   2m = m or m = -1.
   If m = 2, c can be any value, giving lines y = 2x + c.
   If m = -1, c must be 0, giving the line y = -x.

Question 38

Example: Find the line of invariant points and invariant lines for the matrix [[2, 1], [2, 3]].

Answer 38

1. Line of invariant points: y = -x.
2. Invariant lines: y = 2x + c (any c) and y = -x.

Question 39

Why is identifying invariant points and lines useful in transformations?

Answer 39

They provide insights into the geometric effects of transformations, such as rotations, reflections, or stretches.