Find the Inverse [[10,-2,5],[6,-1,4],[1,0,1]]

Problem

[[10,−2,5],[6,−1,4],[1,0,1]]

Solution

1. Calculate the determinant of the matrix using expansion along the third row.

|A|=1*(−8−(−5))−0*(40−30)+1*(−10−(−12))

|A|=1*(−3)+1*(2)

|A|=−1

2. Find the matrix of minors by calculating the determinant of the 2×2 matrix remaining after removing the row and column of each element.

M=[[−1,2,1],[−2,5,2],[−3,10,2]]

3. Apply the cofactor sign pattern to the matrix of minors to create the cofactor matrix.

C=[[−1,−2,1],[2,5,−2],[−3,−10,2]]

4. Transpose the cofactor matrix to find the adjugate matrix.

adj(A)=[[−1,2,−3],[−2,5,−10],[1,−2,2]]

5. Divide by the determinant to find the inverse matrix.

A(−1)=1/(−1)*[[−1,2,−3],[−2,5,−10],[1,−2,2]]

A(−1)=[[1,−2,3],[2,−5,10],[−1,2,−2]]

Final Answer

[[10,−2,5],[6,−1,4],[1,0,1]](−1)=[[1,−2,3],[2,−5,10],[−1,2,−2]]

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