Find the Rank [[-6,2,-4],[9,-8,1],[-9,1,6]]

Problem

rank(−6)

Solution

1. Set up the matrix A to perform row reduction to row-echelon form.

A=[[−6,2,−4],[9,−8,1],[−9,1,6]]

2. Simplify the first row by dividing by −2 to create a leading value.

(R_1)→−1/2*(R_1)

[[3,−1,2],[9,−8,1],[−9,1,6]]

3. Eliminate the entries below the first pivot in the first column.

(R_2)→(R_2)−3*(R_1)

(R_3)→(R_3)+3*(R_1)

[[3,−1,2],[0,−5,−5],[0,−2,12]]

4. Simplify the second row by dividing by −5

(R_2)→−1/5*(R_2)

[[3,−1,2],[0,1,1],[0,−2,12]]

5. Eliminate the entry below the second pivot in the second column.

(R_3)→(R_3)+2*(R_2)

[[3,−1,2],[0,1,1],[0,0,14]]

6. Identify the number of non-zero rows in the row-echelon form.

Number of non-zero rows=3

Final Answer

rank(−6)=3

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