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

Problem

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

Solution

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

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

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

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

det(A)=(3)*(3)−(4)*(5)

det(A)=9−20

det(A)=−11

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

adj(A)=[[3,−4],[−5,3]]

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

A(−1)=1/(−11)*[[3,−4],[−5,3]]

A(−1)=[[−3/11,4/11],[5/11,−3/11]]

Final Answer

[[3,4],[5,3]](−1)=[[−3/11,4/11],[5/11,−3/11]]

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