Find the Inverse [[-6,-9,-8],[2,9,6],[0,1,-1]]

Problem

[[−6,−9,−8],[2,9,6],[0,1,−1]]

Solution

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

det(A)=−6*((9)*(−1)−(6)*(1))−2*((−9)*(−1)−(−8)*(1))+0

det(A)=−6*(−9−6)−2*(9+8)

det(A)=−6*(−15)−2*(17)

det(A)=90−34=56

2. Find the matrix of minors by calculating the determinant of the 2×2 matrix remaining when the row and column of each element are removed.

(M_11)=(9)*(−1)−(6)*(1)=−15

(M_12)=(2)*(−1)−(6)*(0)=−2

(M_13)=(2)*(1)−(9)*(0)=2

(M_21)=(−9)*(−1)−(−8)*(1)=17

(M_22)=(−6)*(−1)−(−8)*(0)=6

(M_23)=(−6)*(1)−(−9)*(0)=−6

(M_31)=(−9)*(6)−(−8)*(9)=18

(M_32)=(−6)*(6)−(−8)*(2)=−20

(M_33)=(−6)*(9)−(−9)*(2)=−36

3. Apply the cofactor signs using the pattern +−+ −+− +−+ to the matrix of minors.

C=[[−15,2,2],[−17,6,6],[18,20,−36]]

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

adj(A)=[[−15,−17,18],[2,6,20],[2,6,−36]]

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

A(−1)=1/56*[[−15,−17,18],[2,6,20],[2,6,−36]]

Final Answer

[[−6,−9,−8],[2,9,6],[0,1,−1]](−1)=[[−15/56,−17/56,9/28],[1/28,3/28,5/14],[1/28,3/28,−9/14]]

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