Find the Inverse [[f,1,0],[1,f,1],[0,1,f]]

Problem

[[ƒ,1,0],[1,ƒ,1],[0,1,ƒ]]

Solution

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

det(A)=ƒ*(ƒ2−1)−1*(ƒ−0)+0

det(A)=ƒ3−ƒ−ƒ

det(A)=ƒ3−2*ƒ

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_11)=ƒ2−1,(M_12)=ƒ,(M_13)=1

(M_21)=ƒ,(M_22)=ƒ2,(M_23)=ƒ

(M_31)=1,(M_32)=ƒ,(M_33)=ƒ2−1

3. Apply the checkerboard of signs to create the matrix of cofactors (C_i*j)=(−1)(i+j)*(M_i*j)

C=[[ƒ2−1,−ƒ,1],[−ƒ,ƒ2,−ƒ],[1,−ƒ,ƒ2−1]]

4. Transpose the cofactor matrix to find the adjugate matrix adj(A) Since this matrix is symmetric, the transpose is identical.

adj(A)=[[ƒ2−1,−ƒ,1],[−ƒ,ƒ2,−ƒ],[1,−ƒ,ƒ2−1]]

5. Divide by the determinant to obtain the inverse matrix A(−1)=1/det(A)*adj(A)

A(−1)=1/(ƒ3−2*ƒ)*[[ƒ2−1,−ƒ,1],[−ƒ,ƒ2,−ƒ],[1,−ƒ,ƒ2−1]]

Final Answer

[[ƒ,1,0],[1,ƒ,1],[0,1,ƒ]](−1)=[[(ƒ2−1)/(ƒ3−2*ƒ),(−1)/(ƒ2−2),1/(ƒ3−2*ƒ)],[(−1)/(ƒ2−2),ƒ/(ƒ2−2),(−1)/(ƒ2−2)],[1/(ƒ3−2*ƒ),(−1)/(ƒ2−2),(ƒ2−1)/(ƒ3−2*ƒ)]]

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