Find the Inverse [[13,8,15],[6,0,7],[7,9,8]]

Problem

[[13,8,15],[6,0,7],[7,9,8]](−1)

Solution

1. Calculate the determinant of the matrix A using the cofactor expansion along the second row.

|A|=−6*(8⋅8−15⋅9)+0−7*(13⋅9−8⋅7)

|A|=−6*(64−135)−7*(117−56)

|A|=−6*(−71)−7*(61)

|A|=426−427=−1

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)=0*(8)−7*(9)=−63

(M_12)=6*(8)−7*(7)=−1

(M_13)=6*(9)−0*(7)=54

(M_21)=8*(8)−15*(9)=−71

(M_22)=13*(8)−15*(7)=−1

(M_23)=13*(9)−8*(7)=61

(M_31)=8*(7)−15*(0)=56

(M_32)=13*(7)−15*(6)=1

(M_33)=13*(0)−8*(6)=−48

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

C=[[−63,1,54],[71,−1,−61],[56,−1,−48]]

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

adj(A)=[[−63,71,56],[1,−1,−1],[54,−61,−48]]

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

A(−1)=1/(−1)*[[−63,71,56],[1,−1,−1],[54,−61,−48]]

A(−1)=[[63,−71,−56],[−1,1,1],[−54,61,48]]

Final Answer

[[13,8,15],[6,0,7],[7,9,8]](−1)=[[63,−71,−56],[−1,1,1],[−54,61,48]]

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