Find the Rank [[1,-7],[2,-10],[3,-13],[4,-16]]

Problem

rank(1)

Solution

1. Identify the matrix A and its dimensions.

A=[[1,−7],[2,−10],[3,−13],[4,−16]]

2. Perform row operations to reach row echelon form. Subtract 2 times the first row from the second row ((R_2)←(R_2)−2*(R_1).

[[1,−7],[0,4],[3,−13],[4,−16]]

3. Perform row operations to eliminate the first element of the third row. Subtract 3 times the first row from the third row ((R_3)←(R_3)−3*(R_1).

[[1,−7],[0,4],[0,8],[4,−16]]

4. Perform row operations to eliminate the first element of the fourth row. Subtract 4 times the first row from the fourth row ((R_4)←(R_4)−4*(R_1).

[[1,−7],[0,4],[0,8],[0,12]]

5. Simplify the remaining rows. Subtract 2 times the second row from the third row ((R_3)←(R_3)−2*(R_2) and 3 times the second row from the fourth row ((R_4)←(R_4)−3*(R_2).

[[1,−7],[0,4],[0,0],[0,0]]

6. Determine the rank by counting the number of non-zero rows in the row echelon form. There are 2 non-zero rows.

rank(A)=2

Final Answer

rank(1)=2

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