Find the Inverse A*B=TR[[23,-16],[-9,7]]

Problem

A⋅B=[[23,−16],[−9,7]]

Solution

1. Identify the matrix for which the inverse needs to be found. Let M=A⋅B

M=[[23,−16],[−9,7]]

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

det(M)=(23)*(7)−(−16)*(−9)

3. Simplify the arithmetic for the determinant.

det(M)=161−144

det(M)=17

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

M(−1)=1/17*[[7,16],[9,23]]

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

M(−1)=[[7/17,16/17],[9/17,23/17]]

Final Answer

(A⋅B)(−1)=[[7/17,16/17],[9/17,23/17]]

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