$6\mathrm{.}6$ Matrix multiplication plays an important role in a number of applications. Two matrices can only be multiplied if the number of columns of the first matrix is equal to the number of rows in the second.

Let's assume we have an $mn$ matrix A and we want to multiply it by an $np$ matrix $B.$ We can express their product as an $mp$ matrix denoted by $AB($or $A*B).$ If we assign $C=AB,$ and $c_{ij}$ denotes the entry in $C$ at position $(i,j),$ then for each element i and $j$ with $1im$ and $1jp{c}_{i,j}={\displaystyle \underset{k=1}{\overset{n}{}}}{a}_{i,k}{b}_{k,j}.$ Now we want to see if we can parallelize the computation of $C.$ Assume that matrices are laid out in memory sequentially as follows: $a_{1,1},a_{2,1},a_{3,1},a_{4,1},$dots, etc.

$6\mathrm{.}6\mathrm{.}1[10]$$$6\mathrm{.}5>$ Assume that we are going to compute $C$ on both a single$-$core shared$-$memory machine and a four$-$core shared$-$memory machine. Compute the speed$-$up we would expect to obtain on the four$-$core machine, ignoring any memory issues.