A is a 4 5 matrix What are the possible ranks of AAT and AT

A is a 4 × 5 matrix. What are the possible ranks of AA^T and A^TA? Why?

Solution

In the matrix product of (AA^T),

rank(AA^T) rank(A) 4.

[ since Rank (AB) min {Rank(A) , Rank(B)}.]

and Rank (A) = Rank (A^T) 4.

hence Rank(AA^T) have possibilites are 0,1,2,3,4.

similarly Rank (A^TA) also have same possibilities.