Determine the geometric meaning of the operators A B and C a

Determine the geometric meaning of the operators A, B, and C acting on R^2, if they are represented by the following matrices: A = (-1 0 0 -1), B = (1 0 0 -1), C = 1/squareroot 2(1 -1 1 1).

Solution

A acting on vector [x y]^T gives [-x -y]

So geometrically it represents reflection in the origin

B acting on vector [x y]^T gives [x -y]

So geometrically it represents reflection in the x axis

C is clock wise rotation by 45 degrees