Find the Rank [[3,1,0],[2,1,-4],[2,-1,20]]

Problem

rank(3)

Solution

1. Set up the matrix A representing the given system.

A=[[3,1,0],[2,1,−4],[2,−1,20]]

2. Perform row operations to create a leading 1 in the first row by subtracting the second row from the first ((R_1)→(R_1)−(R_2).

[[1,0,4],[2,1,−4],[2,−1,20]]

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

[[1,0,4],[0,1,−12],[0,−1,12]]

4. Eliminate the entry below the second pivot by performing (R_3)→(R_3)+(R_2)

[[1,0,4],[0,1,−12],[0,0,0]]

5. Identify the number of non-zero rows in the row-echelon form. Since there are two non-zero rows, the rank is 2

Final Answer

rank(3)=2

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