Inverses and Determinants
March 4, 2010
1. Find a 2 × 2 matrix A
A2 = I
Such that A 6= I and A 6= −I.
2. Find a 2 × 2 matrix A
A4 = I
but A2 6= ±I. (Hint: think of A as a transformation of the plane)
3. a. Draw the image of the triangle under the linear transformation.1
2 −1
A=
−2
2
2
1
-2
-1
0
1
2
P
-1
-2
b. Find the inverse of the matrix A, and draw the image of the triangle
under A−1 .
c. Compute: the area of the triangle, the area of A(triangle), the area
of A−1 (triangle), the determinant of A, and the determinant of A−1 . These
numbers are related in some way. How?
1
(Hint: the origin stays fixed, and you just have to find out where the vertices go. For
example, under A the vertex (1, −1) goes to
1
1 · 0 + (−1) · 1
−1
A
=
=
−1
1 · (−1) + 1 · 0
−1
once you know where all the vertices go, you know where the triangle goes)
Inverses and Determinants
March 4, 2010
4. Do the same thing with the following matrix
2 −1
B=
4 −2
Show that B isn’t injective (one to one) by finding two vectors v1 6= v2
such that Bv1 = Bv2 .
5. Compute the determinant of


1 1 2
A= 1 0 1 
0 1 2
b. Find its inverse A−1 .
c. Find the determinant of its inverse