Find the Inverse

Problem

[[1/√(,5),−14/√(,205)],[2/√(,5),−3/√(,205)]]

Solution

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

A=[[a,b],[c,d]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

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

det(A)=(1/√(,5))*(−3/√(,205))−(−14/√(,205))*(2/√(,5))

det(A)=−3/√(,1025)+28/√(,1025)

det(A)=25/√(,1025)

3. Simplify the determinant using √(,1025)=√(,25×41)=5√(,41)

det(A)=25/(5√(,41))

det(A)=5/√(,41)

4. Apply the inverse formula by swapping the main diagonal elements and negating the off-diagonal elements.

A(−1)=√(,41)/5*[[−3/√(,205),14/√(,205)],[−2/√(,5),1/√(,5)]]

5. Distribute the scalar factor √(,41)/5 into the matrix, noting that √(,205)=√(,41)×√(,5)

A(−1)=[[√(,41)/5*(−3/(√(,41)√(,5))),√(,41)/5*(14/(√(,41)√(,5)))],[√(,41)/5*(−2/√(,5)),√(,41)/5*(1/√(,5))]]

A(−1)=[[−3/(5√(,5)),14/(5√(,5))],[−(2√(,41))/(5√(,5)),√(,41)/(5√(,5))]]

Final Answer

[[1/√(,5),−14/√(,205)],[2/√(,5),−3/√(,205)]](−1)=[[−3/(5√(,5)),14/(5√(,5))],[−(2√(,41))/(5√(,5)),√(,41)/(5√(,5))]]

Want more problems? Check here!