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

Problem

det(1)

Solution

1. Identify the matrix and the method for calculating the determinant. We will use the expansion by minors along the first row.

2. Set up the expansion using the elements of the first row: 1 2 and 1

3. Calculate the first term by multiplying the first element by the determinant of its 2×2 minor:

1⋅det(1)=1⋅(1⋅1−2⋅2)

1⋅(1−4)=−3

4. Calculate the second term, remembering to negate it based on the checkerboard pattern of signs:

−2⋅det(−2)=−2⋅(−2⋅1−2⋅1)

−2⋅(−2−2)=−2⋅(−4)=8

5. Calculate the third term by multiplying the third element by the determinant of its 2×2 minor:

1⋅det(−2)=1⋅(−2⋅2−1⋅1)

1⋅(−4−1)=−5

6. Sum the results of the three terms to find the final determinant:

−3+8−5=0

Final Answer

det(1)=0

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