Download Question 1. Suppose that D : M n×n(F) → F is an alternating, n

Question 1. Suppose that D : Mn×n (F) → F is an alternating, n-linear function. For A ∈ Mn×n (F), let Ai
be the i-th column of A.
(i) For each i = 1, . . . , n, use the multilinearity of D to show

A11
···
A1,i−1
 ..
..
.
..
 .
.

Aj−1,1 · · · Aj−1,i−1
n
X

···
0
D(A) =
Aj,i D 
 0
Aj+1,1 · · · Aj+1,i−1
j=1

 .
..
..
 ..
.
.
An,1
···
An,i−1
that
0
..
.
A1,i+1
..
.
0 Aj−1,i+1
1
0
0 Aj+1,i+1
..
..
.
.
0 An,i+1
···
..
.
···
···
···
..
.
···

A1,n
.. 
. 

Aj−1,n 

0 

Aj+1,n 

.. 
. 
An,n
(ii) Show that

A11
 ..
 .

Aj−1,1

D
 0
Aj+1,1

 .
 ..
An,1
···
..
.
···
···
···
..
.
Aj−1,i−1
0
Aj+1,i−1
..
.
···
An,i−1
A1,i−1
..
.
0
..
.
···
..
.
···
···
···
..
.
A1,i+1
..
.
0 Aj−1,i+1
1
0
0 Aj+1,i+1
..
..
.
.
0 An,i+1
···


A1,n
.. 
. 

Aj−1,n 

0 

Aj+1,n 

.. 
. 
An,n
A11
 ..
 .

Aj−1,1
i+j
= (−1) D(I) det 
Aj+1,1

 .
 ..
An,1
(Hint: For any n × n matrix B, define

B11
 ..
 .

Bj−1,1

E(B) = D 
 0
 Bj,1

 .
 ..
Bn,1
···
..
.
···
···
···
..
.
Bj−1,i−1
0
Bj,i−1
..
.
···
Bn,i−1
B1,i−1
..
.
0
..
.
···
..
.
···
···
..
.
A1,i−1
..
.
A1,i+1
..
.
Aj−1,i−1
Aj+1,i−1
..
.
Aj−1,i+1
Aj+1,i+1
..
.
···
..
.
···
···
..
.
···
An,i−1
An,i+1
···
B1,i
..
.
0 Bj−1,i
1
0
0
Bj,i
..
..
.
.
0 Bn,i
···
..
.
···
···
···
..
.
···

B1,n
.. 
. 

Bj−1,n 

0 

Bj,n 

.. 
. 
Bn,n
and show that B is an alternating, n-linear function. Deduce that E(B) = E(I) det(B).)
(iii) Conclude, for any alternating, n-linear function D : Mn×n (F) → F, that
D(A) = D(I)
n
X
(−1)i+j Ai,j det(Ãi,j )
j=1
and in particular that
det(A) =
n
X
(−1)i+j Ai,j det(Ãi,j ).
j=1
1

A1,n
.. 
. 

Aj−1,n 

Aj+1,n 

.. 
. 
An,n
Question 2. For any n×n matrix A with coefficients in F, let A0 be the matrix where A0ij = (−1)i+j det Ãij .
Prove that
At A0 = det(A)In .
Use this to deduce that A is invertible if and only if det(A) is an invertible element of F.
Question 3. If A is a square matrix and n is an integer that is ≥ 0, let An be the result of multiplying A
with itself n times. By convention, A0 is the identity matrix.
The Fibonacci numbers are defined by the following recursive formula: F0 = F1 = 1 and Fn = Fn−1 +Fn−2
for all n ≥ 2.
Prove that
n Fn+1
1 1
1
=
Fn
1 0
1
for all integers n ≥ 0.
2