Add [[-16,14,-6],[26,-22,12],[-11,10,-6]]^-1

Problem

[[−16,14,−6],[26,−22,12],[−11,10,−6]](−1)

Solution

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

det(A)=−16*((−22)*(−6)−(12)*(10))−14*((26)*(−6)−(12)*(−11))−6*((26)*(10)−(−22)*(−11))

det(A)=−16*(132−120)−14*(−156+132)−6*(260−242)

det(A)=−16*(12)−14*(−24)−6*(18)

det(A)=−192+336−108=36

2. Find the matrix of cofactors by calculating the minor of each element and applying the sign pattern.

(C_11)=+((−22)*(−6)−(12)*(10))=12

(C_12)=−((26)*(−6)−(12)*(−11))=24

(C_13)=+((26)*(10)−(−22)*(−11))=18

(C_21)=−((14)*(−6)−(−6)*(10))=24

(C_22)=+((−16)*(−6)−(−6)*(−11))=30

(C_23)=−((−16)*(10)−(14)*(−11))=6

(C_31)=+((14)*(12)−(−6)*(−22))=36

(C_32)=−((−16)*(12)−(−6)*(26))=36

(C_33)=+((−16)*(−22)−(14)*(26))=−12

3. Form the adjugate matrix by taking the transpose of the cofactor matrix.

adj(A)=[[12,24,36],[24,30,36],[18,6,−12]]

4. Multiply by the reciprocal of the determinant to find the inverse.

A(−1)=1/36*[[12,24,36],[24,30,36],[18,6,−12]]

5. Simplify each element in the matrix.

A(−1)=[[1/3,2/3,1],[2/3,5/6,1],[1/2,1/6,−1/3]]

Final Answer

[[−16,14,−6],[26,−22,12],[−11,10,−6]](−1)=[[1/3,2/3,1],[2/3,5/6,1],[1/2,1/6,−1/3]]

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