Find the Inverse [[7,2,4],[6,6,3],[8,4,9]]

Problem

[[7,2,4],[6,6,3],[8,4,9]](−1)

Solution

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

|A|=7*(6⋅9−3⋅4)−2*(6⋅9−3⋅8)+4*(6⋅4−6⋅8)

|A|=7*(54−12)−2*(54−24)+4*(24−48)

|A|=7*(42)−2*(30)+4*(−24)

|A|=294−60−96=138

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)=42,(M_12)=30,(M_13)=−24

(M_21)=2,(M_22)=31,(M_23)=12

(M_31)=−18,(M_32)=−3,(M_33)=30

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

C=[[42,−30,−24],[−2,31,−12],[−18,3,30]]

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

adj(A)=[[42,−2,−18],[−30,31,3],[−24,−12,30]]

5. Multiply by the reciprocal of the determinant using the formula A(−1)=1/|A|*adj(A)

A(−1)=1/138*[[42,−2,−18],[−30,31,3],[−24,−12,30]]

Final Answer

[[7,2,4],[6,6,3],[8,4,9]](−1)=[[7/23,−1/69,−3/23],[−5/23,31/138,1/46],[−4/23,−2/23,5/23]]

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