4 Assume a
2
+b
2
= 1. One of the three transformations is a rotation,
the other is a reflection about a line, the third is an orthogonal
projection onto a line. Which is which? Find the inverse in the
case of rotation and reflection.
a) C =
a b
b −a
. b) A =
a −b
b a
. c) B =
a
2
ab
ab b
2
. d) For a
2 × 2 matrix A =
a b
c d
, the determinant is defined as det A =
ad − bc. Find the determinant of a shear, of a rotation, of a
reflection about a line, of reflection in the origin, a projection onto
the x axis, a rotation-dilation matrix with parameters (a, b).
Solution:
a) Reflection,
b) Rotation,
c) Orthogonal projection,
d) Shears and rotations have determinant 1. Reflections about a line
have determinant −1. Plane reflections about a point have deter-
minant 1 since they are equivalent to 180
◦
rotations. The rotation-
dilation matrix with parameters (a, b) has determinant a
2
+ b
2
.
5 The matrix multiplication introduced in the next lecture gives
the entry ij of AB as the dot product of the i’th row of A with
j’th column of B. Verify that the product of a reflection dilation
a b
b −a
with an other reflection dilation
c d
d −c
is a rotation
dilation. What is the rotation angle and what is the scaling factor?
Discuss this with somebody else to make sure you understand this
geometrically.
3
