If A is 8 times 5 matrix what is the largest and smallest po

If A is 8 times 5 matrix, what is the largest and smallest possible rank of A? What is the largest possible dimension of Nul A?

Solution

rank (A)=dim col(A)=dim row(A)

there are 5 columns and 8 rows

So , dim(col(A))<=5

Hence, rank<=5

So largest possible rank is 5

A can be a 0 matrix then rank would be 0. So smallest possible rank is 0

rank+dim Nul A=5 =(number of columns of A)

dim Nul A=5- rank A

Hence, dim Nul A can take smallest value of 0 and largest value of 5