Find the Inverse [[2,3,3],[3,2,3],[0,1,4]]

Problem

[[2,3,3],[3,2,3],[0,1,4]](−1)

Solution

1. Calculate the determinant of the matrix A using cofactor expansion along the first column.

det(A)=2*(2⋅4−3⋅1)−3*(3⋅4−3⋅1)+0*(3⋅3−3⋅2)

det(A)=2*(5)−3*(9)

det(A)=10−27=−17

2. Find the matrix of minors by calculating the determinant of the 2×2 matrix remaining when the row and column of each element are removed.

M=[[5,12,3],[9,8,2],[3,−3,−5]]

3. Apply the cofactor signs using the pattern of alternating signs to create the cofactor matrix C

C=[[5,−12,3],[−9,8,−2],[3,3,−5]]

4. Transpose the cofactor matrix to find the adjugate matrix adj(A)

adj(A)=[[5,−9,3],[−12,8,3],[3,−2,−5]]

5. Multiply by the reciprocal of the determinant to find the inverse matrix A(−1)=1/det(A)*adj(A)

A(−1)=1/(−17)*[[5,−9,3],[−12,8,3],[3,−2,−5]]

Final Answer

[[2,3,3],[3,2,3],[0,1,4]](−1)=[[−5/17,9/17,−3/17],[12/17,−8/17,−3/17],[−3/17,2/17,5/17]]

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