1Let A be an mn matrix of rank m Prove that the rows of A fo

1.Let A be an m*n matrix of rank m. Prove that the rows of A form a basis of the row space.

2. Prove that an n*n matrix A is nonsingular if and only if A^t is .

I am confused with those two questions. Please help me out..Thank you so much

Solution

1. rank m=dim row A=dim col A

If rank A=m

Hence, dim row A =m

But, A has m columns so the columns are linearly independent and hence form basis for row space

2.

Let, A be non singular

det(A)=det(A^T)

But, A is non singular so,

det(A) is non zero

Hence, det(A^T) is non zero and hence, A^T is non singular

Let, A^T be non singular

det(A)=det(A^T)

But, A^T is non singular so,

det(A^T) is non zero

Hence, det(A) is non zero and hence, A is non singular