$($a$)$ Under which conditions a matrix could be factorized using the Cholesky

decomposition method. $[2$ marks$]$

$($b$)($i$)$ Follow the codes below:

$($ m $<-$ matrix$($c$(5,1,1,3),2,2))$

$($ cm $<-$ chol$($m$))$

#crossprod$($cm$)=$t$($cm$)\%*\%$ cm #$--=\text{'}$m$\text{'}$

t$($cm$)\%*\%$ cm

crossprod$($cm$)$ #$--=\text{'}$m$\text{'}$

Now,

x $<-$ matrix$($c$(1$:$5,(1$:$5)^2),5,2)$

x $<-$ cbind$($x$,$ x$[,1]+3*$x$[,2])$

colnames$($x$)<-$ letters$[20$:$22]$

m $<-$ crossprod$($x$)$

Give the output matrix m and its properties. $[8$ marks$]$

$($ii$)$ chol$($A$)$ is the built$-$in$-$function to decompose a matrix A in R$.$

Further to $1($b$)($i$),$ what is your expected output by typing chol$($m$)+-$