Find the Inverse [[2,5,9],[5,4,7],[8,6,4]]

Problem

[[2,5,9],[5,4,7],[8,6,4]]

Solution

1. Calculate the determinant of the matrix A using the first row.

det(A)=2*(4⋅4−7⋅6)−5*(5⋅4−7⋅8)+9*(5⋅6−4⋅8)

det(A)=2*(16−42)−5*(20−56)+9*(30−32)

det(A)=2*(−26)−5*(−36)+9*(−2)

det(A)=−52+180−18=110

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)=−26,(M_12)=−36,(M_13)=−2

(M_21)=−34,(M_22)=−64,(M_23)=−28

(M_31)=−1,(M_32)=−31,(M_33)=−17

3. Apply the checkerboard of signs to create the matrix of cofactors C

C=[[−26,36,−2],[34,−64,28],[−1,31,−17]]

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

adj(A)=[[−26,34,−1],[36,−64,31],[−2,28,−17]]

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

A(−1)=1/110*[[−26,34,−1],[36,−64,31],[−2,28,−17]]

Final Answer

[[2,5,9],[5,4,7],[8,6,4]](−1)=[[−13/55,17/55,−1/110],[18/55,−32/55,31/110],[−1/55,14/55,−17/110]]

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