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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 $B A V=0$.
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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 $B A V=0$.
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