Find the Rank A=[[1,-2,0,4],[3,1,1,0],[-1,-5,-1,8]]

Problem

A=[[1,−2,0,4],[3,1,1,0],[−1,−5,−1,8]]

Solution

1. Set up the matrix for row reduction to find the number of non-zero rows in row-echelon form.

A=[[1,−2,0,4],[3,1,1,0],[−1,−5,−1,8]]

2. Eliminate the first entries of the second and third rows using the first row. Perform (R_2)→(R_2)−3*(R_1) and (R_3)→(R_3)+(R_1)

[[1,−2,0,4],[0,7,1,−12],[0,−7,−1,12]]

3. Eliminate the second entry of the third row using the second row. Perform (R_3)→(R_3)+(R_2)

[[1,−2,0,4],[0,7,1,−12],[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)=2

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