Find the Inverse [[5,10],[10,30]]

Problem

[[5,10],[10,30]]

Solution

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

A=[[a,b],[c,d]]=[[5,10],[10,30]]

A(−1)=1/det(A)*[[d,−b],[−c,a]]

2. Calculate the determinant of the matrix.

det(A)=(5)*(30)−(10)*(10)

det(A)=150−100=50

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

adj(A)=[[30,−10],[−10,5]]

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

A(−1)=1/50*[[30,−10],[−10,5]]

5. Simplify each element in the matrix.

A(−1)=[[30/50,(−10)/50],[(−10)/50,5/50]]

A(−1)=[[3/5,−1/5],[−1/5,1/10]]

Final Answer

[[5,10],[10,30]](−1)=[[3/5,−1/5],[−1/5,1/10]]

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