If the subspace of all solutions of Ax 0 as a basis consist

If the subspace of all solutions of Ax = 0 as a basis consisting of three vectors and if A is a 7 times 9 matrix, what is the rank of A? rank A = 2 (Type a whole number.)

Solution

If A is a 7x9 matrix, then Ax =0 is a system of 6 linear equations in 9 variables.Further, if the solution space of this system of equations has a basis consisting of 3 vectors, then it means that there are 3 free variables amongst the 9 given variables. Also, we know that the number of free variables plus the number of leading variables = n, the number of columns of A. Then, there are 6 leading variables so that the rank of A is 6. The answer is Rank A = 6.