Give three different but equivalent definitions of the rank

Give three different (but equivalent) definitions of the rank of an m × n matrix

Solution

1> all minors of order (r+1) are zero but there is atleast one non zero minor of order \'r\' if exist it is called the rank of the matrix

2> The rank of the matrix A is the dimension of the row space of A, and is denoted rank(A)

3> The maximum number of linearly independent rows in a matrix A is called the row rank of A, and the maximum number of linarly independent columns in A is called the column rank of A