2022 Aug 07 | Yiming Zhang

Let $A$ and $B$ be two real $3 \times 3$ matrices such that $\operatorname{rank}(A B)=1$. Which of the following numbers cannot be a value for $\operatorname{rank}(B A)$?

ANSWER: rank(BA) cannot be 3.

Using the following two facts.

$\operatorname{rank}(A B) \leq \min (\operatorname{rank}(A), \operatorname{rank}(B))$ if $B$ is $n \times k$
$\operatorname{rank}(A B)=\operatorname{rank}(A)$ if $B$ is $n \times k$ of rank $n$

By property 1, the only way rank(BA) = 3 is rank(A) = rank(B) = 3. In this case, rank(AB) = rank(A) = 3 by property 2, which contradicts the rank(AB) = 1.

Say you flip a coin 100 times, and see heads 60 times. What is the $99 \%$ confidence interval around your sample mean? (Note: code heads as 1 , standard Normal z-index for a $99 \%$ confidence interval is $2.58$, round to two decimal digits)

Denote $X_i$ be the outcome of the $i-th$ flip, then

$$
\begin{aligned}
\hat{p}=\frac{1}{n} \sum_{i=1}^{n} X_{i}\\
\operatorname{Var}[\hat{p}]=\frac{p(1-p)}{n}
\end{aligned}
$$

since each flip follows a Bernoulli distribution and has a variance of $p(1-p)$. By central limit theorem, $\hat{p}$ has a normal distribution,

$$
\hat{p}\sim\mathcal{N}(p,\sqrt{\frac{p(1-p)}{n}})
$$

What is the area bounded by the functions $f(x)=\sin (2 x)$ and $g(x)=\sin (-2 x)$ for the interval $-\pi \leq x \leq \pi$ ?