Homework 3
YOUR NAME
Instructions: You can either print out the homework and write your answer on it(recommended),
or other ways that clearly shows your work and computations. Write your UNI in the right top
corner of each page. and your name in the left top corner. For computational excercise, circle the
final answer. This homework is due at 4:30pm June, 8th.
Problem 1. Compute the following determinant by anyway you like.
(1)
(2)
1
1
1
4
2
4
0
2
2
1
2
2
4
1902 2345
2
4
⇡
cos 2
Date: Printed on:June 4, 2015;
1
Homework 3
2
(3)
(4)
5
1
1
1
Your UNI:
1
2
4
1
2
1
2
1
1
1
1
1
1
1
2
1
4
2
YOUR NAME
Your UNI:
(5)
1 2
8 9
(6)
9 0
3 9
(7)
2
6
0
0
3
1
0
0
0
0
1
5
0
0
2
7
3
Homework 3
(8)
(9)
0
0
1
5
0
2
0
1
Your UNI:
0
0
2
6
0
5
0
2
3
1
0
0
6
3
0
0
3
0
9
0
6
0
5
0
(Hint: try to switch some row?)
4
YOUR NAME
(10)
2
6
5
7
Your UNI:
3
1
6
9
0
0
1
5
0
0
2
7
Problem 2.
(1) Suppose we know det(A) = 2, what is det(A
1 )?
(2) Suppose we know det(I A) = 3, det(3I
Expand (3I A)(I A))
5
A) = 5, what is det(A2
Why is that?
4A + 3I)? (Hint:
Homework 3
Your UNI:
Problem 3. Find the inverse by method of computing the adjugate of the following matrices
0
1
1
3 5
0 2 A
(1) @ 1
4
1 2
(2)
✓
0
(3) @
a b
c d
1
◆
1
1 1 A
1
6
YOUR NAME
Your UNI:
Problem 4. Find the inverse by method of elementary row transformation of the following matrices
0
1
1
2 1
0 1 A
(1) @ 9
5
1 2
(2)
✓
0
2 3
5 9
1
B 2
(3) B
@ 5
1
◆
1
2
0
9
1
2
1
7
1
0
0 C
C
2 A
3
Problem 5. In this exercise, we prove a fun fact that det(I AB) = det(I BA)
Suppose A is m ⇥ n matrix, B is n ⇥ m matrix.
by using the block matrix row and column transformation.
✓
◆ ✓
◆
Im
Am⇥n
Im AB
(1) Find a block matrix P, such that P
=
Bn⇥m
In
B
In
7
Homework 3
Your UNI:
(2) Find a block matrix Q, such that
(3) Show that det(Im
✓
AB) = det(In
Im
Am⇥n
Bn⇥m
In
◆
Q=
BA)
(4) Calculate the determinant of the matrix:
0
1
1
1
1
1
0
1
1
1
1
1
0
1
1
8
1
1
1
0
1
1
1
1
1
0
✓
Im
Bn⇥m In
BA
◆
YOUR NAME
0
B
B
(Hint: what is B
B
@
1
1
1
1
1
1
C
C
C
C
A
Your UNI:
1 1 1 1 1 ? )
Problem 6. In this exercise, we talk about the construction of complex numbers.
(1) Suppose a is a real number, show that a2
0 (Discuss case by case: a  0, a
0)
(2) Althogh an element of square -1 can not happen
in ◆
real numbers, but it could happen in 2 ⇥ 2
✓
1
matrices over R, Now if we denote J =
, check that J 2 = I2
1
(3) The
✓ complex
◆ number a + bi means the 2 ⇥ 2 matrix over R, which is a + bi = aI + bJ =
a b
. Proof the rule (a + bi)(c + di) = (ac bd) + (ad + bc)i
b a
9
Homework 3
Your UNI:
p
(4) The absolute value of a complex number a + bi is defined to be |a + bi| = det(aI + bJ),
show that this definition is coincide with the absolute value for real numbers. And it saitisfies |a + bi||c + di| = |(a + bi)(c + di)|
(5) Show that if the absolute value of complex number is not equal to 0, then a + bi has recipa bi
rocal, the reciprocal is |a+bi|
2
10