Find the Rank

Problem

rank(0)

Solution

1. Represent the matrix A and swap rows to place a leading 1 in the first row. Swap (R_1) and (R_4)

A=[[1,0,0,−1,3],[2,1,0,3,2],[0,−1,2,1,0],[0,3,−2,0,6],[1,3,−5,8,3]]

2. Eliminate the first column entries below the pivot. Perform (R_2)→(R_2)−2*(R_1) and (R_5)→(R_5)−(R_1)

[[1,0,0,−1,3],[0,1,0,5,−4],[0,−1,2,1,0],[0,3,−2,0,6],[0,3,−5,9,0]]

3. Eliminate the second column entries below the pivot in (R_2) Perform (R_3)→(R_3)+(R_2) (R_4)→(R_4)−3*(R_2) and (R_5)→(R_5)−3*(R_2)

[[1,0,0,−1,3],[0,1,0,5,−4],[0,0,2,6,−4],[0,0,−2,−15,18],[0,0,−5,−6,12]]

4. Normalize the third row by performing (R_3)→1/2*(R_3)

[[1,0,0,−1,3],[0,1,0,5,−4],[0,0,1,3,−2],[0,0,−2,−15,18],[0,0,−5,−6,12]]

5. Eliminate the third column entries below the pivot in (R_3) Perform (R_4)→(R_4)+2*(R_3) and (R_5)→(R_5)+5*(R_3)

[[1,0,0,−1,3],[0,1,0,5,−4],[0,0,1,3,−2],[0,0,0,−9,14],[0,0,0,9,2]]

6. Eliminate the fourth column entry below the pivot in (R_4) Perform (R_5)→(R_5)+(R_4)

[[1,0,0,−1,3],[0,1,0,5,−4],[0,0,1,3,−2],[0,0,0,−9,14],[0,0,0,0,16]]

7. Identify the number of non-zero rows in the row echelon form. There are 5 non-zero rows.

Final Answer

rank(0)=5

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