$2.$ Consider the matrix $\backslash (\backslash$mathbf$\{$B$\}\backslash )$ given by

$\backslash [$

$\backslash$mathbf$\{$B$\}=\backslash$left$[\backslash$begin$\{$array$\}\{$rrr$\}$

$4$ & $7$ & $-1\backslash \backslash$

$-6$ & $-7$ & $4\backslash \backslash$

$6$ & $8$ & $-3$

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

$\backslash ]$

$($a$)$ Find the column$-$sum norm and the Frobenius norm of $\backslash (\backslash$mathbf$\{$B$\}\backslash )($by hand$).$

$($b$)$ Using Matlab, find the eigenvalues of $\backslash (\backslash$mathbf$\{$B$\}^\{$T$\}\backslash$mathbf$\{$B$\}\backslash ),$ and thus determine the spectral norm of $\backslash (\backslash$mathbf$\{$B$\}\backslash )$ by hand. As part of your answer include a write$-$up or print$-$out of the commands used in Matlab as well as the resulting output.

$($c$)$ Using Matlab, find $\backslash (\backslash$mathbf$\{$B$\}^\{-1\}\backslash ),$ as well as its spectral norm. As part of your answer include a write$-$up or print$-$out of the commands used in Matlab as well as the resulting output.

$($d$)$ Using the appropriate result from $($c$),$ find the condition number of $\backslash (\backslash$mathbf$\{$B$\}\backslash )($by hand$)$ based on the column$-$sum norm.

$($e$)$ Using the appropriate results from $($b$)$ and $($c$),$ find the condition number of $\backslash (\backslash$mathbf$\{$B$\}\backslash )($by hand$),$ based on the spectral norm. Verify this result by using the 'cond' command in Matlab. As part of your answer include a write$-$up or print$-$out of the commands used in Matlab as well as the resulting output.