Find the Inverse [[10,9],[-6,-5]]

Problem

[[10,9],[−6,−5]]

Solution

1. Identify the matrix A and its elements.

A=[[a,b],[c,d]]=[[10,9],[−6,−5]]

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

det(A)=(10)*(−5)−(9)*(−6)

det(A)=−50−(−54)

det(A)=4

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

A(−1)=1/4*[[−5,−9],[6,10]]

4. Distribute the scalar 1/4 to each element inside the matrix.

A(−1)=[[−5/4,−9/4],[6/4,10/4]]

5. Simplify the fractions where possible.

A(−1)=[[−1.25,−2.25],[1.5,2.5]]

Final Answer

[[10,9],[−6,−5]](−1)=[[−1.25,−2.25],[1.5,2.5]]

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