Find the Inverse [[5,4],[3,5]]

Problem

[[5,4],[3,5]]

Solution

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

A=[[a,b],[c,d]]=[[5,4],[3,5]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

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

det(A)=(5)*(5)−(4)*(3)

det(A)=25−12

det(A)=13

3. Swap the main diagonal elements and negate the off-diagonal elements to form the adjugate matrix.

adj(A)=[[5,−4],[−3,5]]

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

A(−1)=1/13*[[5,−4],[−3,5]]

Final Answer

[[5,4],[3,5]](−1)=[[5/13,−4/13],[−3/13,5/13]]

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