Compute Rank Of Matrix $a_{ij} = i+j$

Algebra Level 3

Consider the $10$ -by- $10$ matrix $M$ for which the entry in the $i^\text{th}$ row and $j^\text{th}$ column is $i+j$ .

What is the rank of $M?$

The answer is 2.

1 solution

Otto Bretscher
Apr 28, 2016

The first two (independent) columns $C_1$ and $C_2$ span the column space since $C_j=(2-j)C_1+(j-1)C_2$ for $j>2$ . Thus the rank (which is the dimension of the column space) is $\boxed{2}$ .

Compute Rank Of Matrix a i j = i + j a_{ij} = i+j a i j ​ = i + j

The answer is 2.

1 solution

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Compute Rank Of Matrix $a_{ij} = i+j$