What is the rank of a 4 times 5 matrix whose null space is t

What is the rank of a 4 times 5 matrix whose null space is three-dimensional ? If A is a 6 times 7 matrix what is the largest possible dimension of Col(A) ? If P is a 5 times 5 matrix and Nul(P) is the zero subspace, what can you say about solutions of equations of the form Px = b for b R^5?

Solution

As per the rank-nullity theorem, for a m x n matrix A, the rank of (A) + null(A) = n. Here, n = 5 and null(A) = 3. Therefore, rank(A) = 5-3 = 2. Since a matrix has at least one non-zero element, its rank must be greater than zero, i.e. the minimum rank is 1. Then, as per the rank-nullity theorem, the maximum value of Col(A) is 7 – rank(A) = 7 -1 = 6. If Null(P) is the zero subspace, then PX = 0 has only trivial solution, so that P is row equivalent to the 5 × 5 identity matrix I5. Then the equation PX = b , for b R5 will always have a unique solution.