Matrix transformations

Given

prove by mathematical induction that, for n

Given

describe fully the single geometrical transformation P represented by the matrix B

The transformation Q is represented by the matrix Aⁿ

The transformation P followed by the transformation Q is the transformation R, which is represented by the matrix C

Determine C in terms of n

Given that, for a particular value of n, the transformation R maps the point with coordinates (27, 1) to the point with coordinates (a, a), where a is a constant,

determine the matrix that represents the transformation Q

Prove by mathematical induction that, for n

Triangle T has vertices A, B and C.

Triangle T is transformed to triangle T′ by the transformation represented by Mⁿ where n

Given that

• triangle T has an area of 5 cm²

• triangle T′ has an area of 1215 cm²

• vertex A (2, –2) is transformed to vertex A ′(123, –2)

determine

the value of n

the value of a

Prove by mathematical induction that, for n

Triangle T has vertices A, B and C.

Triangle T is transformed to triangle T′ by the transformation represented by Mⁿ where n

Given that

• triangle T has an area of 5 cm²

• triangle T′ has an area of 1215 cm²

• vertex A (2, –2) is transformed to vertex A ′(123, –2)

determine

the value of n

the value of a

The transformation P is an enlargement, centre the origin, with scale factor k, where k 0

The transformation Q is a rotation through angle θ degrees anticlockwise about the origin.

The transformation P followed by the transformation Q is represented by the matrix

Determine

the value of k,

the smallest value of θ

A square S has vertices at the points with coordinates (0, 0), (a, –a), (2a, 0) and (a, a) where a is a constant.

The square S is transformed to the square S′ by the transformation represented by M.

Determine, in terms of a, the area of S′

The transformation P is an enlargement, centre the origin, with scale factor k, where k 0

The transformation Q is a rotation through angle θ degrees anticlockwise about the origin.

The transformation P followed by the transformation Q is represented by the matrix

Determine

the value of k,

the smallest value of θ

A square S has vertices at the points with coordinates (0, 0), (a, –a), (2a, 0) and (a, a) where a is a constant.

The square S is transformed to the square S′ by the transformation represented by M.

Determine, in terms of a, the area of S′

The matrix A represents the linear transformation M.

Prove that, for the linear transformation M, there are no invariant lines.

The matrix A represents the linear transformation M.

Prove that, for the linear transformation M, there are no invariant lines.