Suppose that P is a 4×5 matrix such that every solution of the equation Px = 0…
2021
Suppose that P is a 4×5 matrix such that every solution of the equation Px = 0 is a scalar multiple of [2 5 4 3 1]T. The rank of P is __________ .
Answer: 4 — Concept: For an m×n matrix A, the rank–nullity theorem states rank(A) + nullity(A) = n, where the nullity is the dimension of the null space — the solution…
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Correct answer: 4
Concept: For an m×n matrix A, the rank–nullity theorem states rank(A) + nullity(A) = n, where the nullity is the dimension of the null space — the solution space of Ax = 0.
Application:
P is a 4×5 matrix, so the number of columns is n = 5.
Every solution of Px = 0 is a scalar multiple of the single nonzero vector [2 5 4 3 1]T, so the null space is exactly the span of that one vector — a 1-dimensional space. Hence nullity(P) = 1.
By the rank–nullity theorem: rank(P) = n − nullity(P) = 5 − 1 = 4.
Cross-check: Since P has only 4 rows, rank(P) ≤ min(4, 5) = 4. The computed rank of 4 exactly meets this upper bound, meaning P has full row rank — consistent with the given nullity of 1.
Answer: 4