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$(2$ points$)$ Consider the following table of term frequencies for a corpus consisting of $3$ documents denoted by Doc$1,$ Doc$2,$ Doc$3.$ Compute the term$-$frequency invere$-$document$-$frequency $($TF$-$IDF$)$ matrix for the terms "car", "auto", "insurance", and "best".

$\backslash$begin$\{$tabular$\}\{$c$|$ccc$\}$

$\backslash$hline terms & Doc$1$ & Doc$2$ & Doc$3\backslash \backslash$

$\backslash$hline car & $27$ & $4$ & $24\backslash \backslash$

auto & $3$ & $33$ & $0\backslash \backslash$

insurance & $0$ & $33$ & $29\backslash \backslash$

best & $14$ & $0$ & $17\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$($a$)$ Compute the inverse$-$document$-$frequency $($IDF$)$ matrix:

$\backslash [$

$\backslash$mathrm$\{$IDF$\}=\backslash$left$[\backslash$begin$\{$array$\}\{$c$|\}$

$\backslash$hline $\backslash \backslash$

$\backslash$hline $\backslash \backslash$

$\backslash$hline $\backslash$log $(3/2)\backslash \backslash$

$\backslash$hline $\backslash$log $(3/2)\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$array$\}\backslash$right$]$

$\backslash ]$

$($b$)$ Compute the TF$-$IDF matrix: