Find the Inverse [[1,2],[7,16]]

Problem

[[1,2],[7,16]]

Solution

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

A=[[a,b],[c,d]]=[[1,2],[7,16]]

A(−1)=1/det(A)*[[d,−b],[−c,a]]

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

det(A)=(1)*(16)−(2)*(7)

det(A)=16−14

det(A)=2

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

adj(A)=[[16,−2],[−7,1]]

4. Multiply the adjugate matrix by the scalar 1/det(A)

A(−1)=1/2*[[16,−2],[−7,1]]

A(−1)=[[8,−1],[−3.5,0.5]]

Final Answer

[[1,2],[7,16]](−1)=[[8,−1],[−3.5,0.5]]

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