The rows and columns of determinants have equal rights, that is,

.

Proof: The property follows from the definition of determinants. Really, is the sum of the same terms as . The only difference between them is the order of the terms that does not influence the value of the sum.

Property 2

brief

Multiplying any row or column of a determinant by a number , multiplies the determinant by that number:

This means that the common factor of a row (or column) can be taken outside the sign of a determinant.

Proof: Every term of the sum

contains one and only one element of a given row or column of the determinant. If the row (or column) is multiplied by a number, each the term is multiplied by that number.

Property 3

brief

Interchanging any two rows (or columns) of a matrix changes the sign of the determinant:

Proof: By Theorem 1, any transposition changes the inversion parity of a given permutation. Therefore, each term of the sum

changes its sign.

Property 4

brief

If a determinant has a zero-row or zero-column then the determinant is equal to zero:

Proof: Every product of the sum

contains a zero factor and so equals zero.

Property 5

brief

If a determinant has two rows (or columns) equal then the determinant is equal to zero:

Proof: If the two identical rows (or columns) are interchanged, then by Property 3 the determinant changes its sign. On the other hand, the rows (or columns) are equal and so the determinant must keep its value:

Property 6

brief

If two rows (or columns) of a determinant are proportional to each other then the determinant is equal to zero:

Proof: Multiplying the i-th row of the determinant by the constant of proportionality we obtain the determinant with equal rows.

Property 7

brief

If each element of a row (or column) of a determinant is the sum of two items then

Proof:

Property 8

brief

A determinant holds its value if a row (column) multiplied by a number is added to another one:

Proof: The determinant on the right side hand can be represented as the sum of two determinant, one of which contains two rows proportional and so equals zero.

Property 10

brief

The determinant of a triangular matrix is equal to the product of the diagonal elements:

In particular, the determinant of an identity matrix is equal to unity.

Proof: There is only a single non-zero element a1,1 on the first column. Then we have to ignore the first row and chose a non-zero element on the second column that is a2,2. Likewise, on the third column we can take only the element a3,3to get a non-zero product of elements, and so on. Therefore, all appropriate permutations of indexes give a zero yield to the determinant except the product of the diagonal elements.