Find the Determinant [[2,-1,2],[12,1,-3],[6,-1,6]]

Problem

det(2)

Solution

1. Apply the cofactor expansion along the first row to break the 3×3 determinant into 2×2 determinants.

2. Calculate the first term by multiplying the element (a_11)=2 by the determinant of the remaining 2×2 matrix.

2⋅det(1)=2*(1⋅6−(−3)⋅(−1))

2*(6−3)=6

3. Calculate the second term by multiplying the element (a_12)=−1 by its cofactor (noting the sign change for position 1,2.

−(−1)⋅det(12)=1*(12⋅6−(−3)⋅6)

1*(72+18)=90

4. Calculate the third term by multiplying the element (a_13)=2 by the determinant of the remaining 2×2 matrix.

2⋅det(12)=2*(12⋅(−1)−1⋅6)

2*(−12−6)=−36

5. Sum the results of the three terms to find the final determinant value.

6+90−36=60

Final Answer

det(2)=60

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