An 2016 times 1015 matrix A has the property that for some c

An 2016 times 1015 matrix A has the property that for some column b, the system AX = b has exactly one solution. What is the rank of A?

Solution

Let A be a 2016x 1015 matrix. Since the system Ax = b has a unique ( exactly one) solution for some column b, therefore, the reduced row echelon form of A has a pivot position in all its rows. Also, then all the columns of A are linearly independent > Therefore, the rank of A being the dimension of the vector space generated (or spanned) by its columns is 1015.