## Transcribed Text

Problem 1. Compute the following determinant by anyway you like.
(1)
140
122
1 4 2
(2)
1 22
2 44
1902 2345 ⇡
cos 2
1
(3)
2
5
1
2
4
1
2
1
2
1
2
1
4
(4)
1
1
1 1
1
1 1
1
1
(5)
1 2
8 9
(6)
9 0
3 9
(7)
2 3
6 1
0 0
0 0
0 0
0 0
1 2
5 7
(8)
0 0
0 0
3 6
1 3
1 2
5 6
0 0
0 0
(9)
0 0
2 5
3 6
0 0
0 0
1 2
9 5
0 0
(Hint: try to switch some row?)
4
(10)
2 3
6 1
0 0
0 0
5 6
7 9
1 2
5 7
Problem 2.
(1) Suppose we know det(A) = 2, what is det(A1)? Why is that?
(2) Suppose we know det(I A) = 3, det(3I A) = 5, what is det(A2 4A + 3I)? (Hint:
Expand (3I A)(I A))
0
m i
1
Proble 3. Find the verse by method of computing the adjugate of the following matrices n
(1) @
1 3 5
1 0 2
4 1 2 A
(2) ✓ a b
c d
(3)
0
@
1
1 1
1
1
A
Problem 4. Find the inverse by method of elementary row transformation of the following matrices
(1)
0
@
1 21
9 01
5 1 2
1
A
(2) ✓ 2 3
5 9
(3)
0
BB@
1
2
5
1
1
2
0
9
1
2
1
7
0
0
2
3
1
CCA
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.
(1) Find a block matrix P, such that P
✓ Im Am⇥n
Bn⇥m In
=
✓ Im AB
B In
(2) Find a block matrix Q, such that ✓ Im Am⇥n
Bn⇥m In
Q =
✓ Im
Bn⇥m In BA
(3) Show that det(Im AB) = det(In BA)
(4) Calculate the determinant of the matrix:
0
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
0
1
1
0
1
1
8
(Hint: what is
0
BBBB@
1
1
1
1
1
1
CCCCA
11111
? )
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
matrices over R, Now if we denote J =
✓ 1
1
, check that J2 = I2
(3) The complex number
✓
a + bi means the 2 ⇥ 2 matrix over R, which is a + bi = aI + bJ =
a b
b a
. Proof the rule (a + bi)(c + di)=(ac bd)+(ad + bc)i
9
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 reciprocal, the reciprocal is a
a
+
bi
bi
| |2

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.