Find the Inverse [[2,-8],[11,11]]

Problem

[[2,−8],[11,11]]

Solution

1. Identify the matrix A and the formula for the inverse of a 2×2 matrix.

A=[[a,b],[c,d]]=[[2,−8],[11,11]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

2. Calculate the determinant of the matrix using the formula det(A)=a*d−b*c

det(A)=(2)*(11)−(−8)*(11)

det(A)=22+88

det(A)=110

3. Construct the adjugate matrix by swapping the main diagonal elements and changing the signs of the off-diagonal elements.

adj(A)=[[11,8],[−11,2]]

4. Multiply the adjugate matrix by the reciprocal of the determinant.

A(−1)=1/110*[[11,8],[−11,2]]

5. Simplify the resulting matrix by distributing the scalar fraction to each element.

A(−1)=[[11/110,8/110],[(−11)/110,2/110]]

A(−1)=[[1/10,4/55],[−1/10,1/55]]

Final Answer

[[2,−8],[11,11]](−1)=[[1/10,4/55],[−1/10,1/55]]

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