91影视

We have given two$n脳n$matrices A and B whose entries are positive or zero.

Since all entries of A are less than or equal to 鈥�s鈥�, then we have

$鈬�a_{ip}鈮�s$ (1)

Since all column sums of B are less than or equal to 鈥�谤鈥�, then we have

$鈬�\underset{p=1}{\overset{n}{鈭�}}{b}_{pj}鈮�r$ (2)

Now we have to show that all entries of the matrix AB are less than or equal to 鈥�s谤鈥�.

i.e. $\underset{p=1}{\overset{n}{鈭�}}{a}_{ip}{b}_{pj}鈮�sr$

Since we have from equation (1) and (2),

localid="1659346125962" $a_{ip}鈮�s\mathrm{and}\underset{p=1}{\overset{n}{鈭�}}{b}_{pj}鈮�r$

Now,

$$\begin{gathered} \underset{p=1}{\overset{n}{鈭�}}{a}_{ip}{b}_{pj}鈮�\underset{p=1}{\overset{n}{鈭�}}S.b_{pj}(鈭�a_{ip}鈮�s) \\ 鈬�\underset{p=1}{\overset{n}{鈭�}}{a}_{ip}{b}_{pj}鈮�s\underset{p=1}{\overset{n}{鈭�}}{b}_{pj} \\ 鈬�\underset{p=1}{\overset{n}{鈭�}}{a}_{ip}{b}_{pj}鈮�sr\left(鈭�\underset{p=1}{\overset{n}{鈭�}}{b}_{pj}鈮�r\right) \end{gathered}$$

This shows that all entries of the matrix AB are less than or equal to 鈥�s谤鈥�.

If we have two $n脳n$matrices A and B whose entries are positive or zero and all entries of A are less than or equal to 鈥�s鈥�, and all column sums of B are less than or equal to 鈥�谤鈥� (the column sum of a matrix is the sum of all the entries in its column) then all entries of the matrix AB are less than or equal to 鈥�sr鈥�.