Answer to Question #101030 in Linear Algebra for John Yeo

Question

1. Find the largest possible value for the rank of A and the smallest possible value for the nullity of A.
A is 8 × 8

a. The largest possible value for the rank of A is ?
b. The smallest possible value for the nullity of A is ?

2. Find the largest possible value for the rank of A and the smallest possible value for the nullity of A.
A is 3 × 8

a. The largest possible value for the rank of A is ?
b. The smallest possible value for the nullity of A is ?

Answer 1

We use the following theorem: dimension of rows of A=rank A+nullity A

If $A\in M_{n\times m}(\mathbb R)$, then maximal rank of A is $\min\{n,m\}$. Since dimension of rows of A is $m$, we have that minimal nullity of A is $m-\min\{m,n\}$

1)$m=n=8$, so maximal rank of A is $\min\{8,8\}=8$, minimal nullity of A is $8-\min\{8,8\}=0$

2)$n=3, m=8$, so maximal rank of A is $\min\{3,8\}=3$, minimal nullity of A is $8-\min\{3,8\}=5$

Answer: 1a)8, 1b)0, 2a)3, 2b)5