Example 5 Find the sum of all numbers in the following square matrix
\begin{tabular}{ccccc}
1901 & 1902 & $\cdots$ & 1949 & 1950 \\
1902 & 1903 & $\cdots$ & 1950 & 1951 \\
$\cdots$ & & & & \\
1950 & 1951 & $\cdots$ & 1998 & 1999
\end{tabular}
Many people first use formula (5) to find the sum of the first row:
$$
S_{1}=\frac{1901+1950}{2} \times 50=96275 \text {; }
$$

Then they find the sums of the 2nd, 3rd, ..., 50th rows; finally, they add these sums together. If one notices that each number in the second row is 1 greater than the corresponding number in the previous row, then the sums of the rows are:
$$
\begin{array}{l}
S_{2}=96275+50, \\
S_{3}=96275+50 \times 2, \\
\cdots \cdots \\
S_{50}=96275+50 \times 49 .
\end{array}
$$

Thus, one can use (5) again to find $S_{1}+S_{2}+\cdots+S_{50}$. However, a simpler approach is to apply the idea of averaging. By averaging each number with the number symmetrically opposite to it about the center, each average is $\frac{1901+1999}{2}=1950$.
Thus, the total sum is
$$
1950 \times 50 \times 50=4875000 \text {. }
$$

It is evident that even a simple and common example contains a lot of content, and it is not the case that a single abstraction can strip it of all its essence and spirit.