Answer to Question #91406 in Linear Algebra for Sajid

Question

Choose the correct answer.
Q. Let A be an m×n matrix, then which of the following is true.
a. Rank(A)≤m, Nulity(A)≥n
b. Rank(A)≤min(m,n), Nulity(A)≥n-min(m,n)
c. Rank(A)≤m, Nulity(A)≥n-max(m,n)
d. Rank(A)≤m, Nulity(A) ≤n-max(m,n )

Answer 1

Due to the rank and nullity theorem:

$$\mathrm{Rank}(A) + \mathrm{Nullity}(A) = n$$

for any m×n matrix A. The rank obeys

$$\mathrm{Rank}(A) \leq \min(m,n) \leq m$$

where < if n is the minimal one and = in the other case.

However,

$$\mathrm{Nullity}(A) = n - \mathrm{Rank}(A) \geq n - \min(m,n) \geq n - m$$

Therefore the correct answer is b) Rank(A)≤min(m,n), Nulity(A)≥n-min(m,n)