Simplify the Matrix [[3,-12,-6],[2,-8,-4],[4,-16,-8]]

Problem

[[3,−12,−6],[2,−8,−4],[4,−16,−8]]

Solution

1. Identify the relationship between the columns of the matrix.

2. Observe that every column is a scalar multiple of the first column.

3. Factor out common constants from each row to see the underlying structure.

4. Notice that the second column is −4 times the first column, and the third column is −2 times the first column.

5. Perform row reduction (Gaussian elimination) to find the row echelon form.

6. Subtract 2/3 of the first row from the second row: (R_2)→(R_2)−2/3*(R_1)

7. Subtract 4/3 of the first row from the third row: (R_3)→(R_3)−4/3*(R_1)

8. Simplify the resulting matrix, which reveals that the second and third rows become zero.

Final Answer

[[3,−12,−6],[2,−8,−4],[4,−16,−8]]=[[1,−4,−2],[0,0,0],[0,0,0]]

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