Find the Inverse

Problem

[[1/11,1/11,−5/11],[1/11,1/11,1/22],[1/11,−10/11,1/22]]

Solution

1. Identify the matrix A and the formula for the inverse of a 3×3 matrix.

A=[[1/11,1/11,−5/11],[1/11,1/11,1/22],[1/11,−10/11,1/22]]

A(−1)=1/det(A)*adj(A)

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

det(A)=1/11*(1/11⋅1/22−1/22⋅(−10)/11)−1/11*(1/11⋅1/22−1/22⋅1/11)+(−5)/11*(1/11⋅(−10)/11−1/11⋅1/11)

det(A)=1/11*(1/242+10/242)−1/11*(0)−5/11*(−10/121−1/121)

det(A)=1/11⋅11/242−5/11⋅(−11)/121

det(A)=1/242+5/121=1/242+10/242=11/242=1/22

3. Find the matrix of cofactors C by calculating the minor of each element.

(C_11)=1/11⋅1/22−1/22⋅(−10)/11=11/242=1/22

(C_12)=−(1/11⋅1/22−1/22⋅1/11)=0

(C_13)=1/11⋅(−10)/11−1/11⋅1/11=−11/121=−1/11

(C_21)=−(1/11⋅1/22−(−5)/11⋅(−10)/11)=−(1/242−50/121)=−((1−100)/242)=99/242=9/22

(C_22)=1/11⋅1/22−(−5)/11⋅1/11=1/242+5/121=11/242=1/22

(C_23)=−(1/11⋅(−10)/11−1/11⋅1/11)=−(−11/121)=1/11

(C_31)=1/11⋅1/22−(−5)/11⋅1/11=11/242=1/22

(C_32)=−(1/11⋅1/22−(−5)/11⋅1/11)=−11/242=−1/22

(C_33)=1/11⋅1/11−1/11⋅1/11=0

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

adj(A)=[[1/22,9/22,1/22],[0,1/22,−1/22],[−1/11,1/11,0]]

5. Multiply the adjugate matrix by 1/det(A)=22

A(−1)=22*[[1/22,9/22,1/22],[0,1/22,−1/22],[−1/11,1/11,0]]

A(−1)=[[1,9,1],[0,1,−1],[−2,2,0]]

Final Answer

[[1/11,1/11,−5/11],[1/11,1/11,1/22],[1/11,−10/11,1/22]](−1)=[[1,9,1],[0,1,−1],[−2,2,0]]

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