Find the Inverse [[2,6,9],[-1,1,3],[4,4,0]]

Problem

[[2,6,9],[−1,1,3],[4,4,0]]

Solution

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

|A|=4*(6⋅3−9⋅1)−4*(2⋅3−9⋅−1)+0

|A|=4*(18−9)−4*(6+9)

|A|=4*(9)−4*(15)

|A|=36−60=−24

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=[[−12,−12,−8],[−36,−36,−16],[9,15,8]]

3. Apply the cofactor sign pattern to the matrix of minors to create the matrix of cofactors C

C=[[−12,12,−8],[36,−36,16],[9,−15,8]]

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

adj(A)=[[−12,36,9],[12,−36,−15],[−8,16,8]]

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

A(−1)=1/(−24)*[[−12,36,9],[12,−36,−15],[−8,16,8]]

6. Simplify the fractions in the resulting matrix.

A(−1)=[[1/2,−3/2,−3/8],[−1/2,3/2,5/8],[1/3,−2/3,−1/3]]

Final Answer

[[2,6,9],[−1,1,3],[4,4,0]](−1)=[[1/2,−3/2,−3/8],[−1/2,3/2,5/8],[1/3,−2/3,−1/3]]

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