Find the Rank [[1,2,3],[4,5,6],[7,8,9]]

Problem

rank(1)

Solution

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

A=[[1,2,3],[4,5,6],[7,8,9]]

2. Eliminate the first column entries below the first pivot by performing the row operations (R_2)→(R_2)−4*(R_1) and (R_3)→(R_3)−7*(R_1)

[[1,2,3],[0,−3,−6],[0,−6,−12]]

3. Simplify the second row by dividing (R_2) by −3 to create a leading 1.

[[1,2,3],[0,1,2],[0,−6,−12]]

4. Eliminate the second column entry below the second pivot by performing the row operation (R_3)→(R_3)+6*(R_2)

[[1,2,3],[0,1,2],[0,0,0]]

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

Final Answer

rank(1)=2

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