Find the Inverse M=[[1,8],[-1,-7]]

Problem

M=[[1,8],[−1,−7]]

Solution

1. Identify the formula for the inverse of a 2×2 matrix M=[[a,b],[c,d]] which is M(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

2. Calculate the determinant det(M)=a*d−b*c using the values a=1 b=8 c=−1 and d=−7

det(M)=(1)*(−7)−(8)*(−1)

det(M)=−7+8

det(M)=1

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

adj(M)=[[−7,−8],[1,1]]

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

M(−1)=1/1*[[−7,−8],[1,1]]

M(−1)=[[−7,−8],[1,1]]

Final Answer

[[1,8],[−1,−7]](−1)=[[−7,−8],[1,1]]

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