III. The most important properties
of Determinants

7. Basic properties of Determinants

Consider first properties of d_n, which follow readily from Definition (6.1).

Since the law of formation (6.1) of |A| applies to every matrix A with n rows and n columns, you can conceive that the elements of A are n² variables and thus find from (6.1) that |A| is a polynomial in these variables. We will express the fact that each term of |A| contains a single factor from each row of A (since there occurs once each of the numbers 1, 2, ··· , n as first subscript) as follows:

I: The determinant |A| is linear and homogeneous with respect to the elements of each row of the matrix A.

Hence follows immmediately

Ia: If all the terms in a row of A are 0, then |A| = 0.

Ib: If you multiply all elements of any row of A by an arbitrary number l, then |A| becomes l|A|.

Moreover, it follows from Ia:

Ic: If all elements of the i-th row of A have the form

a_ik = b_ik + c_ik, (7.1)

that is, are sums of two elements, then

|A| = |B| + |C|, (7.2)

where

In fact, replace in (6.1) each element a_ik by the sum (7.1) and perform the multiplications. If you then collect all the terms with b_ik and all the terms with c_ik, you find

How does |A| change when two of its rows are interchanged? In order to answer this question, we consider the terms

belonging to |A|'s permutation

P = [p₁, ···,p_i,···, p_j,···, p_n].

Denote by A₁ the matrix which arises from A by this operation. Consider |A₁| 's terms belonging to the permutation

P = [p₁, ···, p_i,···, p_j,···, p_n],

which arises out of P by the interchange of the numbers p_i and p_j. This terms is

The products of the a_rj in (7.3) and (7.4) agree with each other, while by Theorem 2 of Section 4

c (p₁, ···, p_i,···, p_j,···, p_n) = - c (p₁, ···, p_j,···, p_i,···, p_n).

Consequently, Every term of |A| has in |A₁| the opposite sign, and obviously, every term in |A₁| has in |A| the opposite sign, whence |A₁| = - |A|. Thus,

II: The determinant |A| changes its sign when two rows in the matrix A are interchanged.

This result yields immediately

IIa: If the matrix A has two equal rows, then |A| = 0.

This occurs because, on the one hand, an interchange of two equal rows does not change |A|, on the other hand, by II, it does change the sign. Hence |A|=-|A|, and therefore |A| = 0.

If you add in the matrix A to the elements of the i-th row l times corresponding elements of another row, say, the j-th row, you obtain the matrix A^* in which the elements in the i-th row have the form

a^*_ik = a_ik + la_jk (i j; k = 1, ···, n).

while all other elements of A^* are the same as in A. Hence you can employ during the evaluation of the determinant |A|^* Rule Ic above and obtain

|A^*| = |A| + |A₀|,

where all but the i-th row of the matrix A₀ agree with the corresponding ones of the matrix A, while the i-th row of A₀ is equal to the j-th row of A multiplied by l, or, what is the same, of A₀. Taking Ib into account, you have

The determinant on the right hand side contains two equal rows. whence, by II, it has the value 0. Therefore |A₀| = 0 and one has

Rule IIb: The determinant |A| does not change its value if an arbitrary multiple of one row is added to another row.

The evaluation of the so called unit matrix

is especially simple. In this matrix a₁₁ = a₂₂ = ··· = a_nn = 1 and a_ik= 0 for ik. If you want to emphasize that E has n rows and n columns, you denote it by E_n. In the sum (6.1) for |E|, only one term is non-zero, indeed the one which belongs to the base sequence of the second subscripts. In all other terms, there appear factors with i p_i which are zero, whence

|E| = 1. (7.6)

The diagonal line from the top on the left hand side to the bottom on the right handside of the matrix A as well as of the determinant |A| is called the principal diagonal.

The diagonal matrix

has the determinant

You use Ib to prove this result. In fact, you can extract from |B| for i=1,2,···,n from the i-th row the factor b_i, when Ib yields

Naturally, the determinant of a matrix with one column and one row - with only one element - is the element itself!

last next