Correlation matrices can often be written as sums of outer products, such as Nv NvT Tp p p pv p$=1$ p$=1$ v$11=,=$ N NR x x C x t where each outer product xp$($xp $)$T $,$ for example, combines two N$-$dimensional vectors to create an N by N matrix. $($a$)$ Give the maximum value of rank$($R$),$ in terms of Nv and N $.($Note that rank$($R$)=$ N if R is nonsingular$).($b$)$ Give the maximum value of rank$($C$),$ in terms of Nv $,$ N $,$ and M$.$