Find the Determinant [[0,-1,a],[3,-a,1],[1,-2,3]]

Problem

det(0)

Solution

1. Apply cofactor expansion along the first row to simplify the calculation since the first element is zero.

2. Calculate the first term by multiplying the element 0 by its minor determinant.

0⋅det(−a)=0

3. Calculate the second term by multiplying the negative of the element −1 by its minor determinant.

−(−1)⋅det(3)=1⋅(3⋅3−1⋅1)

1⋅(9−1)=8

4. Calculate the third term by multiplying the element a by its minor determinant.

a⋅det(3)=a⋅(3⋅(−2)−(−a)⋅1)

a⋅(−6+a)=a2−6*a

5. Sum the results of the cofactor expansion to find the final determinant value.

0+8+a2−6*a=a2−6*a+8

Final Answer

det(0)=a2−6*a+8

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