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

Problem

det(1)

Solution

1. Set up the expansion along the first row to calculate the determinant of the 3×3 matrix.

2. Calculate the first term by multiplying the element (a_11) by the determinant of its 2×2 minor.

1⋅det(−1)=1⋅((−1)*(3)−(2)*(−2))

1⋅(−3+4)=1

3. Calculate the second term by multiplying the element (a_12) by the determinant of its 2×2 minor, noting the sign change in cofactor expansion.

−0⋅det(2)=0

4. Calculate the third term by multiplying the element (a_13) by the determinant of its 2×2 minor.

1⋅det(2)=1⋅((2)*(−2)−(−1)*(1))

1⋅(−4+1)=−3

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

1−0+(−3)=−2

Final Answer

det(1)=−2

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