Find the Determinant [[x,y,z],[2x,-y,-z],[x,2y,-z]]

Problem

det(x)

Solution

1. Set up the determinant calculation using cofactor expansion along the first row.

2. Calculate the first term by multiplying x by the determinant of the 2×2 matrix remaining after removing the first row and first column.

x*det(−y)=x*((−y)*(−z)−(−z)*(2*y))

x*(y*z+2*y*z)=3*x*y*z

3. Calculate the second term by multiplying −y by the determinant of the 2×2 matrix remaining after removing the first row and second column.

−y*det(2*x)=−y*((2*x)*(−z)−(−z)*(x))

−y*(−2*x*z+x*z)=−y*(−x*z)=x*y*z

4. Calculate the third term by multiplying z by the determinant of the 2×2 matrix remaining after removing the first row and third column.

z*det(2*x)=z*((2*x)*(2*y)−(−y)*(x))

z*(4*x*y+x*y)=5*x*y*z

5. Combine the results of the three terms to find the total determinant.

3*x*y*z+x*y*z+5*x*y*z=9*x*y*z

Final Answer

det(x)=9*x*y*z

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