Find the Rank

Problem

rank(1)

Solution

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

A=[[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]

2. Eliminate the first entries of the second, third, and fourth rows using the first row. Perform (R_2)→(R_2)−5*(R_1) (R_3)→(R_3)−9*(R_1) and (R_4)→(R_4)−13*(R_1)

[[1,2,3,4],[0,−4,−8,−12],[0,−8,−16,−24],[0,−12,−24,−36]]

3. Simplify the second row by dividing by −4 ((R_2)→−1/4*(R_2).

[[1,2,3,4],[0,1,2,3],[0,−8,−16,−24],[0,−12,−24,−36]]

4. Eliminate the second entries of the third and fourth rows using the second row. Perform (R_3)→(R_3)+8*(R_2) and (R_4)→(R_4)+12*(R_2)

[[1,2,3,4],[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(1)=2

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