Find the Inverse [[6,-13],[-13,34]]

Problem

[[6,−13],[−13,34]]

Solution

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

A=[[a,b],[c,d]]=[[6,−13],[−13,34]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

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

det(A)=(6)*(34)−(−13)*(−13)

det(A)=204−169

det(A)=35

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

adj(A)=[[34,13],[13,6]]

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

A(−1)=1/35*[[34,13],[13,6]]

Final Answer

[[6,−13],[−13,34]](−1)=[[34/35,13/35],[13/35,6/35]]

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