If X = [ α_ij ], i ≥ 1, j ≤ n, with α_ij = i - j. Then the number of linearly independent Rows are: 1. 0 2. 1 3. n 4. n - 1

Steps to Solve

1. Understanding the matrix structure

The elements of the matrix $X = [\alpha_{ij}]$ are defined as $\alpha_{ij} = i - j$. This implies each element in the $i^{th}$ row and $j^{th}$ column is determined by the difference between the row index and the column index.

2. Examining row dependencies

Consider rows of the matrix $X$. Each row $i$ can be expressed as: $$ R_i = [i - 1, i - 2, i - 3, \ldots, i - n] $$

This shows that each row can be rewritten based on its first element, which decreases linearly as $j$ increases.

3. Generating the first few rows

If we write out a few rows, we get:

• For $i=1$: $[0, -1, -2, \ldots, 1-n]$
• For $i=2$: $[1, 0, -1, \ldots, 2-n]$
• For $i=3$: $[2, 1, 0, \ldots, 3-n]$

4. Identifying linear dependence

Notice that any row can be formed by the previous row plus a fixed vector. For instance: $$ R_{i+1} = R_i + [1, 1, 1, \ldots, 1] $$ This structure leads to linear dependence, and specifically only one row can provide the unique information.

5. Determining the number of linearly independent rows

Since all rows can be derived from each other through this linear combination, the number of linearly independent rows is limited.

Thus, after analyzing the structure, we can conclude that there is only 1 linearly independent row.