Find the Inverse [[22000,45000],[18000,27000]]

Problem

[[22000,45000],[18000,27000]]

Solution

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

A=[[a,b],[c,d]]=[[22000,45000],[18000,27000]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

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

det(A)=(22000)*(27000)−(45000)*(18000)

det(A)=594000000−810000000

det(A)=−216000000

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

adj(A)=[[27000,−45000],[−18000,22000]]

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

A(−1)=1/(−216000000)*[[27000,−45000],[−18000,22000]]

5. Simplify each entry in the matrix by dividing by −216000000

27000/(−216000000)=−27/216000=−1/8000

(−45000)/(−216000000)=45/216000=1/4800

(−18000)/(−216000000)=18/216000=1/12000

22000/(−216000000)=−22/216000=−11/108000

Final Answer

[[22000,45000],[18000,27000]](−1)=[[−1/8000,1/4800],[1/12000,−11/108000]]

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