Find the Rank

Problem

rank(−3)

Solution

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

A=[[−3,6,−1,1,−7],[1,−2,2,3,−1],[2,−4,5,8,−4]]

2. Swap the first and second rows to obtain a leading 1 in the first row.

(R_1)↔(R_2)⇒[[1,−2,2,3,−1],[−3,6,−1,1,−7],[2,−4,5,8,−4]]

3. Eliminate the first entries of the second and third rows using the first row.

(R_2)+3*(R_1)→(R_2)⇒[[1,−2,2,3,−1],[0,0,5,10,−10],[2,−4,5,8,−4]]

(R_3)−2*(R_1)→(R_3)⇒[[1,−2,2,3,−1],[0,0,5,10,−10],[0,0,1,2,−2]]

4. Simplify the second row by dividing all elements by 5

1/5*(R_2)→(R_2)⇒[[1,−2,2,3,−1],[0,0,1,2,−2],[0,0,1,2,−2]]

5. Eliminate the third row by subtracting the second row from it.

(R_3)−(R_2)→(R_3)⇒[[1,−2,2,3,−1],[0,0,1,2,−2],[0,0,0,0,0]]

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

Final Answer

rank(−3)=2

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