Simplify the Matrix [[1,-1,0],[2,2,1],[0,16,-1]]

Problem

[[1,−1,0],[2,2,1],[0,16,−1]]

Solution

1. Identify the task as finding the determinant of the given 3×3 matrix, as "simplify" for a square matrix with constant entries typically refers to evaluating its determinant.

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

3. Calculate the first term by multiplying the element (a_11)=1 by the determinant of the remaining 2×2 matrix.

1⋅det(2)=1⋅(2*(−1)−1*(16))

1⋅(−2−16)=−18

4. Calculate the second term by multiplying the element (a_12)=−1 by −1 (due to the sign pattern) and the determinant of the remaining 2×2 matrix.

−(−1)⋅det(2)=1⋅(2*(−1)−1*(0))

1⋅(−2−0)=−2

5. Calculate the third term by multiplying the element (a_13)=0 by the determinant of the remaining 2×2 matrix.

0⋅det(2)=0

6. Sum the results of the expansions to find the final value.

−18+(−2)+0=−20

Final Answer

det(1)=−20

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