Linear Algebra

312 Chapter 4. Determinants
3.34 [Am. Math. Mon., Jan. 1949] LetSbe the sum of the integer elements of a
magic square of order three and let D be the value of the square considered as a
determinant. Show that D/S is an integer.
3.35 [Am. Math. Mon., Jun. 1931] Show that the determinant of the n 2 elements
in the upper left corner of the Pascal triangle
1 1 1 1 . .
1 2 3 . .
1 3 . .
1
.
.
. .
has the value unity.
4.I.4 Determinants Exist
This subsection is optional. It consists of proofs of two results from the prior
subsection. These proofs involve the properties of permutations, which will not
be used later, except in the optional Jordan Canonical Form subsection.
The prior subsection attacks the problem of showing that for any size there
is a determinant function on the set of square matrices of that size by using
multilinearity to develop the permutation expansion.
�
�
�t1,1
t1,2 �
... t1,n�
�
�t2,1
t2,2 �
... t2,n�
�
� .
� = t
�
� .
�
1,φ1(1)t2,φ1(2) ···tn,φ1(n)|Pφ1 �
�tn,1
tn,2 ... tn,n
�
|
+ t1,φ2(1)t2,φ2(2) ···tn,φ2(n)|Pφ2 |
.
=
+ t 1,φk(1)t 2,φk(2) ···t n,φk(n)|Pφk |
�
permutations φ
t 1,φ(1)t 2,φ(2) ···t n,φ(n) |Pφ|
This reduces the problem to showing that there is a determinant function on
the set of permutation matrices of that size.
Of course, a permutation matrix can be row-swapped to the identity matrix
and to calculate its determinant we can keep track of the number of row swaps.
However, the problem is still not solved. We still have not shown that the result
is well-defined. For instance, the determinant of
⎛ ⎞
0 1 0 0
⎜
Pφ = ⎜1
0 0 0 ⎟
⎝0
0 1 0⎠
0 0 0 1