Find the Inverse of the Resulting Matrix [[1,0],[0,1]][[1,3],[-2,5]]

Problem

([[1,0],[0,1]]*[[1,3],[−2,5]])(−1)

Solution

1. Multiply the matrices. Since the first matrix is the identity matrix I the product is simply the second matrix.

[[1,0],[0,1]]*[[1,3],[−2,5]]=[[1,3],[−2,5]]

2. Calculate the determinant of the resulting matrix A=[[a,b],[c,d]] using the formula det(A)=a*d−b*c

det(A)=(1)*(5)−(3)*(−2)

det(A)=5+6=11

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

A(−1)=1/11*[[5,−3],[2,1]]

4. Distribute the scalar factor into the matrix to find the final form.

A(−1)=[[5/11,−3/11],[2/11,1/11]]

Final Answer

([[1,0],[0,1]]*[[1,3],[−2,5]])(−1)=[[5/11,−3/11],[2/11,1/11]]

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