#### Matrices and Transformation

**VECTORS**

- A vector quantity has both magnitude and direction. For example:
- Vectors a and b represented by the line segments can be added using the parallelogram rule or the nose-to-tail method

- Multiplication by a scalar:
- A scalar quantity has a magnitude but no direction.
- The negative sign reverses the direction of the vector.

- Column vector:
- The top number is the horizontal component and the bottom component is the vertical component.

- Parallel vectors:
- Vectors are parallel if they have the same direction.
- In general, the vector
is parallel to

- Modulus of a vector:
- In general, if

- In general, if

**MATRICES**

- Addition:
- Multiplication by scalar:
- Multiplication by vector:
- You can only multiply if the number of columns on the left matrix is equal to the number of rows on the right matrix.

- Determinant:
- Determinant = leading diagonal – secondary diagonal

- Inverse:
- To work out the inverse of a matrix, alternate the leading diagonal’s elements, negate the secondary diagonal, and multiply by
^{1}/_{determinant}

- To work out the inverse of a matrix, alternate the leading diagonal’s elements, negate the secondary diagonal, and multiply by

**TRANSFORMATION**

- Reflection (M):
- When describing a reflection, the position of the mirror line is essential.

- Rotation (R):
- To describe a rotation, the center of rotation, the angle of rotation, and the direction of the rotation are required.
- A clockwise rotation is negative and an anticlockwise rotation is positive.

- Translation (T):
- When describing a translation, it is necessary to give the translation vector

- Enlargement (E):
- To describe an enlargement, state the scale factor,k, and the center of enlargement.
- If k > 0, both object and image lie on the same side of the center of enlargement.
- If k < 0, object and image lie on opposite sides of the center of enlargement.

**TRANSFORMATION BY MATRICES**

- Reflection:
— Reflection in the x – axis — Reflection in the y – axis — Reflection in the line y = -x

- Enlargement:
— where k = scale factor and center of enlargement = (0, 0)

- Rotation:
— Rotation 90° anticlockwise, center (0, 0) — Rotation 90° clockwise, center (0, 0) — Rotation 180° clockwise/anticlockwise, center (0, 0)