Matrices whose Rows and Columns form an Arithmetic Progression

Problem

Let’s consider the following \(5 \times 5\) matrix of positive integers:

\[ \begin{pmatrix} d & 2d & 3d & ? & 5d \\ 11 & ? & ? & ? & ? \\ ? & 32 & ? & ? & ? \\ ? & ? & ? & ? & ? \\ ? & ? & 79 & ? & ? \\ \end{pmatrix} \\ \]

Check if it can be completed so that in each row and in each column the numbers form an arithmetic progression of positive integers.
Recall that an arithmetic progression of integers with common difference \(d\) is a sequence of this type:

\[ \{a,a+d,a+2d,a+3d, \cdots\} \\ \]

where \(a\) is the first integer of the sequence. For example the following sequence:

\[ \{3,10,17,24,31, \cdots \} \\ \]

is an arithmetic progression with common difference equal to \( 7 \).

Hint
Based on the definition of arithmetic progression, first determine the elements \(a_{22},a_{23}\). Then complete the matrix.

Solution

\[ \begin{pmatrix} 5 & 10 & 15 & 20 & 25 \\ 11 & 21 & 31 & 41 & 51 \\ 17 & 32 & 47 & 62 & 77 \\ 23 & 43 & 63 & 83 & 103 \\ 29 & 54 & 79 & 104 & 129 \\ \end{pmatrix} \]