Properties of Determinants

Property 1

The value of a determinant is unaltered by interchanging its rows and columns.

Property 2

If any two rows (columns) of a determinant are interchanged the determinant changes its sign but its numerical value is unaltered.

Property 3

If two rows (columns) of a determinant are identical then the value of the determinant is zero.

Property 4

If every element in a row (or column) of a determinant is multiplied by a constant k then the value of the determinant is multiplied by k.

If two rows (columns) of a determinant are proportional i.e. one row (column) is a scalar multiple of other row (column) then its value is zero.

Property 5

If every element in any row (column) can be expressed as the sum of two quantities then given determinant can be expressed as the sum of two determinants of the same order with the elements of the remaining rows (columns) of both being the same.

Property 6

A determinant is unaltered when to each element of any row (column) is added to those of several other rows (columns) multiplied respectively by constant factors.