Find the Rank [[3,0,2,2],[-6,42,24,54],[21,-21,0,-15]]

Problem

rank(3)

Solution

1. Write the matrix A and identify the goal to find the number of non-zero rows in its row-echelon form.

A=[[3,0,2,2],[−6,42,24,54],[21,−21,0,−15]]

2. Eliminate the first entries of the second and third rows by performing row operations (R_2)→(R_2)+2*(R_1) and (R_3)→(R_3)−7*(R_1)

[[3,0,2,2],[0,42,28,58],[0,−21,−14,−29]]

3. Eliminate the second entry of the third row by performing the row operation (R_3)→(R_3)+1/2*(R_2)

[[3,0,2,2],[0,42,28,58],[0,0,0,0]]

4. Count the number of non-zero rows in the resulting row-echelon form to determine the rank.

rank(A)=2

Final Answer

rank(3)=2

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