Chapter 4 - Vector Spaces - 4.9 The Rank-Nullity Theorem - True-False Review - Page 330: f

False

Assume that $A=\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ $B=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $are an $m × n$ matrices We can see $rank (A)=rank (B)=1\\ \rightarrow nullity (A)=nullity (B)=1\\ \rightarrow A+B=\begin{bmatrix} 1 & 1\\ 0 & 0 \end{bmatrix}\\ \rightarrow rank(A+B)=1\\ \rightarrow nullity (A+B)=1$ Since $nullity (A+B)=1 \ne 2 =nullity A+nullity B$ Hence, the statement is false.