Find the Inverse of the Resulting Matrix [[1,3,-2],[-1,2,4]][[1,1],[3,2],[-2,4]]

Problem

([[1,3,−2],[−1,2,4]]*[[1,1],[3,2],[−2,4]])(−1)

Solution

1. Multiply the two matrices to find the resulting matrix A

A=[[(1)*(1)+(3)*(3)+(−2)*(−2),(1)*(1)+(3)*(2)+(−2)*(4)],[(−1)*(1)+(2)*(3)+(4)*(−2),(−1)*(1)+(2)*(2)+(4)*(4)]]

2. Simplify the arithmetic for each element of matrix A

A=[[1+9+4,1+6−8],[−1+6−8,−1+4+16]]

A=[[14,−1],[−3,19]]

3. Calculate the determinant of matrix A using the formula det(A)=a*d−b*c

det(A)=(14)*(19)−(−1)*(−3)

det(A)=266−3

det(A)=263

4. Apply the inverse formula for a 2×2 matrix, which is A(−1)=1/det(A)*[[d,−b],[−c,a]]

A(−1)=1/263*[[19,1],[3,14]]

5. Distribute the scalar into the matrix to get the final form.

A(−1)=[[19/263,1/263],[3/263,14/263]]

Final Answer

([[1,3,−2],[−1,2,4]]*[[1,1],[3,2],[−2,4]])(−1)=[[19/263,1/263],[3/263,14/263]]

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