Rank and Nullity of a Matrix

Relevant wiki: subspace

$3^{th}$ row = $- (2^{th} - 2\cdot1^{th})$

and $2^{th}$ and $1^{th}$ rows are linearly independent $\Rightarrow \text{Rank A} = 2$ , and due to $\text{ dim( Ker A) + Rank A = 4 (first theorem for isomorhism in vector spaces)}$ we get $\text{ nullity of A = 2 = dim(Ker A)}$

Let denote, $R_1, R_2, R_3$ be first second and third rows of above given matrix then,

Clearly, $R_3 = 2• R_2 - R_1$ so that, $R_3$ is Linearly dependent on $R_1, R_2$ and one can check $R_1, R_2$ are linearly independent(since they are not multiple of each other!) Hence, $Rank(A)=2$

But by Rank-nullity theorem we know,

$Rank(A) +nullity(A) =$ number of columns of $A$

$→ 2 + nullity(A)= 4$

$→nullity(A)= 2$