Linear Algebra

Section I. Definition 299
1.15 Prove that for 2×2 matrices, the determinant of a matrix equals the determinant
of its transpose. Does that also hold for 3×3 matrices?
� 1.16 Is the determinant function linear — is det(x·T +y·S) =x·det(T )+y·det(S)?
1.17 Show that if A is 3×3 thendet(c · A) =c 3 · det(A) for any scalar c.
1.18 Which real numbers θ make
�cos �
θ − sin θ
sin θ cos θ
singular? Explain geometrically.
1.19 [Am. Math. Mon., Apr. 1955] If a third order determinant has elements 1, 2,
... , 9, what is the maximum value it may have?
4.I.2 Properties of Determinants
As described above, we want a formula to determine whether an n×n matrix
is nonsingular. We will not begin by stating such a formula. Instead, we will
begin by considering the function that such a formula calculates. We will define
the function by its properties, then prove that the function with these properties
exist and is unique and also describe formulas that compute this function.
(Because we will show that the function exists and is unique, from the start we
will say ‘det(T )’ instead of ‘if there is a determinant function then det(T )’ and
‘the determinant’ instead of ‘any determinant’.)
2.1 Definition A n×n determinant is a function det: Mn×n → R such that
(1) det(�ρ1,...,k· �ρi + �ρj,...,�ρn) = det(�ρ1,...,�ρj,...,�ρn) fori �= j
(2) det(�ρ1,... ,�ρj,...,�ρi,...,�ρn) =− det(�ρ1,...,�ρi,...,�ρj,...,�ρn) fori �= j
(3) det(�ρ1,...,k�ρi,...,�ρn) =k · det(�ρ1,...,�ρi,...,�ρn) fork �= 0
(4) det(I) = 1 where I is an identity matrix
(the �ρ ’s are the rows of the matrix). We often write |T | for det(T ).
2.2 Remark Property (2) is redundant since
T ρi+ρj
−→ −ρj+ρi
−→ ρi+ρj
−→ −ρi
−→ ˆ T
swaps rows i and j. It is listed only for convenience.
The first result shows that a function satisfying these conditions gives a
criteria for nonsingularity. (Its last sentence is that, in the context of the first
three conditions, (4) is equivalent to the condition that the determinant of an
echelon form matrix is the product down the diagonal.)