# Let $$A$$ and $$B$$ be $$n \times n$$ matrices with $$A B=0$$. Give a proof or counterexample for each of the following.

461 views
Let $$A$$ and $$B$$ be $$n \times n$$ matrices with $$A B=0$$. Give a proof or counterexample for each of the following.

a) Either $$A=0$$ or $$B=0$$ (or both).
b) $$B A=0$$
c) If $$\operatorname{det} A=-3$$, then $$B=0$$.
d) If $$B$$ is invertible then $$A=0$$.
e) There is a vector $$V \neq 0$$ such that $$B A V=0$$.

a) Proof: If $$A B=0$$, then it follows that every entry of the product matrix is 0 . Hence, either all entries of the first matrix $$A$$ are 0 , or all entries of the second matrix $$B$$ are 0 , or both.

b) Proof: If $$A B=0$$, then we have
$B A = (A B)^T = 0^T = 0$

c) Counterexample: Consider the matrices
$A = \begin{bmatrix}-3 & 0 \\ 0 & -3\end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$

It follows that $$A B=\left[\begin{array}{cc}-3 & 0 \\ 0 & -6\end{array}\right]=0$$, but $$B \equiv 0$$. Hence, the statement is false.
d) Counterexample: Consider the matrices
$A = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$
It follows that $$A B=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]=0$$, and $$B$$ is invertible, but $$A \equiv 0$$. Hence, the statement is false.

e) Proof: If $$A B=0$$, then it follows that for every vector $$V$$, the product $$B A V=0$$. Hence, there exists a non-zero vector $$V$$ such that $$B A V=0$$.