Find the Rank

Problem

rank(0)

Solution

1. Represent the matrix A and identify the rows.

A=[[0,1,2,3],[10,11,12,13],[20,21,22,23],[30,31,32,33]]

2. Perform row operations to create zeros in the first column. Swap (R_1) and (R_2) to get a non-zero pivot.

(R_1)↔(R_2)⇒[[10,11,12,13],[0,1,2,3],[20,21,22,23],[30,31,32,33]]

3. Eliminate the entries below the first pivot in the first column using (R_3)→(R_3)−2*(R_1) and (R_4)→(R_4)−3*(R_1)

[[10,11,12,13],[0,1,2,3],[0,−1,−2,−3],[0,−2,−4,−6]]

4. Eliminate the entries below the second pivot in the second column using (R_3)→(R_3)+(R_2) and (R_4)→(R_4)+2*(R_2)

[[10,11,12,13],[0,1,2,3],[0,0,0,0],[0,0,0,0]]

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

Final Answer

rank(0)=2

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