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

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⋅1)=1

3. Calculate the second term by multiplying the element (a_12) by the determinant of its 2×2 minor, remembering to negate the term according to the cofactor pattern.

−(−2)⋅det(0)=2⋅(0⋅3−2⋅1)=−4

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

1⋅det(0)=1⋅(0⋅1−1⋅1)=−1

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

1+(−4)+(−1)=−4

Final Answer

det(1)=−4

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