Matrix transformations

Write down the 2 × 2 matrix that represents a rotation of 90° anticlockwise about the origin.

Write down the 3 × 3 matrix that represents a reflection in the x – z plane.

Find N² in terms of a.

Find det N in terms of a.

Describe geometrically the single transformation consisting of T_A followed by T_B.

By considering the transformation T_A, determine the matrix A⁴²³.

The transformation T_C is represented by the matrix C, where

The region R is defined by the set of points (x, y) satisfying the inequality x² + y² ⩽ 36.

The region R′ is defined as the image of R under T_C.

Find the exact area of the region R ′.

Sketch the region R ′, specifying all the points where the boundary of R ′ intersects the

coordinate axes.

The points P, Q and R have coordinates (0, 2, 3), (2, 0, 1) and (1, 3, 0) respectively.

The acute angle between the line segments PQ and PR is θ.

The triangle PQR lies in the plane Π.

Determine an equation for Π, giving your answer in the form ax + by + cz = d, where a, b, c

and d are integers.

Determine whether the line with equation y = kx in part (a) is a line of invariant points

under T.

Define the transformation represented by A.

Show that the area of any object shape is invariant under the transformation represented by A.

Find A⁴.

Describe the transformation that A represents.

The matrix B represents a reflection in the plane x = 0.

Write down the matrix B.

The point P has coordinates (2, 3, 4). The point P ′ is the image of P under the transformation represented by B.

Find the coordinates of P ′.