Find the Rank [[2,-1,-1,5],[-1,1,2,-3],[1,-2,3,4]]

Problem

rank(2)

Solution

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

A=[[2,−1,−1,5],[−1,1,2,−3],[1,−2,3,4]]

2. Swap rows to get a leading 1 in the first row. Swap (R_1) and (R_3)

[[1,−2,3,4],[−1,1,2,−3],[2,−1,−1,5]]

3. Eliminate entries below the first pivot. Perform (R_2)→(R_2)+(R_1) and (R_3)→(R_3)−2*(R_1)

[[1,−2,3,4],[0,−1,5,1],[0,3,−7,−3]]

4. Normalize the second row by multiplying (R_2) by −1

[[1,−2,3,4],[0,1,−5,−1],[0,3,−7,−3]]

5. Eliminate the entry below the second pivot. Perform (R_3)→(R_3)−3*(R_2)

[[1,−2,3,4],[0,1,−5,−1],[0,0,8,0]]

6. Identify the number of non-zero rows in the row-echelon form. There are 3 rows with non-zero entries.

Final Answer

rank(2)=3

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