Find the Rank

Problem

Rank*[[1,−3,1,0,−1],[1,−2,1,−1,0],[1,−1,1,−2,1]]

Solution

1. Write the matrix and prepare for row reduction to find the number of non-zero rows in row-echelon form.

A=[[1,−3,1,0,−1],[1,−2,1,−1,0],[1,−1,1,−2,1]]

2. Eliminate the first entry of the second and third rows by subtracting the first row from them ((R_2)→(R_2)−(R_1) and (R_3)→(R_3)−(R_1).

[[1,−3,1,0,−1],[0,1,0,−1,1],[0,2,0,−2,2]]

3. Eliminate the second entry of the third row by subtracting twice the second row from the third row ((R_3)→(R_3)−2*(R_2).

[[1,−3,1,0,−1],[0,1,0,−1,1],[0,0,0,0,0]]

4. Identify the number of non-zero rows in the resulting row-echelon form. There are 2 non-zero rows.

Rank(A)=2

Final Answer

Rank*[[1,−3,1,0,−1],[1,−2,1,−1,0],[1,−1,1,−2,1]]=2

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