let be an matrix

the column rank of is the rank of its column vectors

the row rank of is the rank of its row vectors

, column rank=row rank=1

, column rank=row rank=2

, column rank=row rank=2

has full rank

, not full rank：

has full rank

If are full rank matrices, then so is

in general,

has no immediate dependence on ,, example:

find ranks of and

given , its determinant is

full rank

is the area of the parallelogram surronded the its column (or row) vectors

given , its determinant is

full rank
is the volume of the parallelepiped surronded its column (or row) vectors

let be an matrix, the minor of , denoted by , is the determinant of the matrix obtained by removing the -th row and the -th column in

is then called the co-factor of

, to expand by the first row:

, to expand by the first column:

find following determinants

find following determinants

the determinant of a triangle matrix is the product of its main diagonal elements

the determinant stays invariant upon transpose

the determinant vanishes upon linear dependence of rows/columns

swapping two rows flips the sign of the determinant

scaling one row by also scales the determinant by (attention!)

adding a multiple of one row to anther row leaves the determinant invariant

find

adding a multiple of row 1 to row 2:

adding a multiple of row 1 to row 3:

adding a multiple of row 2 to row 3

taking the product of the main diagonal elements we get , which is also the determinant of the original matrix

find following determinants by using elementary row operations

let be a permutation of

if but ,then is called an inversion (pair), the parity of the permutation is the oddness of its total number of inversions

example： is odd since there are 3 inversions:
example： is even since there are 2 inversions:

where is the set of all permutations of , while is the total number of inversions in

example：

rank and full rank

determinant and its properties

computation of determinants