Exercise 3 .pdf - Math 3A c 2022 March Boedihardjo...

Math 3A Exercise March Boedihardjo c 2022
Important note: The questions below vary a lot in difficulty. For the difficult questions, be patient
when thinking about them. You will learn much more if you spend hours and hours thinking and
get nothing before looking at the solution than if you look at the solution directly without thinking.
Notation: If A is a square matrix then A
k
is the product A . . . A
| {z }
k
. For example, A
2
= AA .
I is the identity matrix.
1. Write down a 2 × 2 matrix A (with real entries) such that A
10
= I and A
k
6 = I for all 1 ≤ k ≤ 9.
2. Find matrices C and D such that C
1 3
2 5
=
2 - 3
1 2
and
1 3
2 5
D =
2 - 3
1 2
.
3. Suppose that A
- 1
=
1 2
3 1
and B
- 1
=
√
2 1
1 3
. Find a vector v such that ABv =
1
0
.
4. Suppose that A =
a b
c 3
is not invertible and A
1
1
=
1
2
. Find a, b, c .
5. Suppose that A is a 2 × 2 invertible matrix such that A
4
5
=
1
1
and A
1
√
3
=
1
0
.
Find A
- 1
1
3
.
6. Suppose that A is a 2 × 2 rotation matrix such that A
2
a
=
1
3
where a > 0. Find a .
7. (i) Show that every 2 × 2 rotation matrix is invertible.
(ii) Show that if A is a 2 × 2 rotation matrix then A +
1
2
I is invertible.
8. Which of the following 2 × 2 matrices are rotation matrices?
(i)
1 0
0 1
(ii)
- 1 0
0 - 1
(iii)
0 - 1
1 0
(iv)
3 - 4
4 3
(v)
1
5
3 - 4
4 3
(vi)
1
5
3 4