$)=(5pt$ Suppose you have a sample of $2000$ students from Syracuse University. Each student bought one tube of toothpaste in February $2023$ and one tube of toothpaste in March $2023.$ They bought brand A$,$ B$,$ C$,$ D$,$ or E$.$ The brand switching matrix is provided below:

$2.($a$)$ How many students purchased D in February $2023?$

Answer:

$2.($b$)$ How many students purchased D in March $2023?$

Answer:

$2.($c$)$ How many students who bought D in February $2023$ switched to other brands in March $2023?($These are people who bought D in February, but bought some other brand in March.$)$

Answer:

$2.($d$)$ How many students did not buy D in February $2023,$ but bought D in March $2023?$

Answer:

$2.($e$)$ Based on the brand$-$switching matrix, which brand or brands, if any, are competitors of brand $D?$

Answer:

Note: A brand $x$ is a competitor of brand $D$ under all of the following conditions:

$Dx>0$ and $xD>0($two$-$way switching between $D$ and $x)$

$Dx>0$ and $xD=0($one$-$way switching from $D$ to $x)$

$Dx=0$ and $xD>0($one$-$way switching from $x$ to $D)$

$x$ is not a competitor of $D$ if and only if $Dx=0$ and $xD=0.$