Finding the nullity of Matrix A (m x n) I have been given a question to find the nullity of A. A is a m x n matrix, what are the possible values of nullity(A)? Values given as options are : a) (m-1) ≤ nullity(A) b) nullity(A) ≥ m c) nullity(A) ≤ n d) nullity(A)=0 And all options seems to be true to me. I am sure about options c and d. But options a) and b) are also holding true for few matrix examples. For example : !nullity is greater than number of rows

> **Rank-Nullity theorem** :
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> If there is a matrix $A$ with $m$ rows and $n$ columns over a field, then $$Rank ( A) + Nullity (A) = n $$
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> Also $Rank(A)\leq min\\{m,n\\}$

Let $n \lt m$, then $Rank(A) \leq n$

**Case I:** If in a matrix $A$, $Rank (A)=n$, then $Nullity (A)=0$

**Case II:** If in a matrix $A$, $0 \leq Rank (A) \lt n$, then $Nullity (A)\leq n$

Let $n \gt m$, then $Rank(A) \leq m$

**Case I:** If in a matrix $A$, $Rank (A)=m$, then $Nullity (A)=n-m$

**Case II:** If in a matrix $A$, $0 \leq Rank (A) \lt m$, then $Nullity (A)\leq n-m$

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Although your given example satisfy both the option $a$ and $b$, but from one particular example you can't conclude that those statements are true for all.