Find the Determinant

Problem

(−2*x*z)*(−2*y*z)*det(−2*x2)

Solution

1. Simplify the scalar coefficient outside the determinant.

(−2*x*z)*(−2*y*z)=4*x*y*z2

2. Factor out common terms from the rows of the matrix to simplify the determinant calculation. Factor −2*x from the first row, −1 from the second row, and −1 from the third row.

det(A)=(−2*x)*(−1)*(−1)*det(x)

3. Perform row operations to create zeros. Replace (R_2) with (R_2)+2*y*(R_1) and (R_3) with (R_3)+2*z*(R_1)

(R_2)→−2*x*y+2*y(x)=0

(R_2)→(1−2*y2)+2*y(y)=1

(R_2)→−2*y*z+2*y(z)=0

(R_3)→−2*x*z+2*z(x)=0

(R_3)→−2*y*z+2*z(y)=0

(R_3)→(1−2*z2)+2*z(z)=1

4. Evaluate the determinant of the resulting upper triangular matrix.

det(x)=x(1)*(1)=x

5. Combine all factors including the initial scalar and the factors from row operations.

Result=(4*x*y*z2)⋅(−2*x)⋅(x)

Result=−8*x3*y*z2

Final Answer

(−2*x*z)*(−2*y*z)*det(−2*x2)=−8*x3*y*z2

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