Suppose that P is a 4 x 5 matrix such that every solution of the equation Px = 0 is a scalar multiple of [2, 4, 3, 1]^T....

Question

Suppose that P is a 4 x 5 matrix such that every solution of the equation Px = 0 is a scalar multiple of [2, 4, 3, 1]^T. The rank of P is __________

Accepted Answer

rank(P) = 4 P is a 4x5 matrix, so it maps R^5 to R^4. The null space of P (all solutions to Px=0) is spanned by a single vector [2,4,3,1]^T. Since the null space has dimension 1 (only scalar multiples of one vector), by the rank-nullity theorem, rank(P) = 5 - 1 = 4.