Consider the matrix

$A=\left[\begin{array}{cccc}5& 7& 6& 5\\ 7& 10& 8& 7\\ 6& 8& 10& 9\\ 5& 7& 9& 10\end{array}\right]$

$($a$)$ Show that $A$ is positive definite. $($b$)$ Use Matlab's chol function to compute the

Cholesky factor $R.$ Use format short. $($c$)$ Using $R,$ find the normalized factors $L$ and

$D.$ Display their entries as rational numbers, then check that $LD{L}^T=A.$

${?}^1$ To time trid$(\{$:$a,b,g),$ for example, use the following script:

s speed.m

tic

for $j=1$:$5$

trid$(a,b,g)$;

end

toc$/5$

Before running the script, suppress all output $($printing$,$ plotting$)$ from trid. Run the

script $10$ times and take the smallest value.