Find the Inverse [[1,2],[3,4]]

Problem

[[1,2],[3,4]]

Solution

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

A=[[a,b],[c,d]]=[[1,2],[3,4]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

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

det(A)=(1)*(4)−(2)*(3)

det(A)=4−6

det(A)=−2

3. Swap the main diagonal elements and negate the off-diagonal elements to form the adjugate matrix.

adj(A)=[[4,−2],[−3,1]]

4. Multiply the adjugate matrix by the reciprocal of the determinant.

A(−1)=1/(−2)*[[4,−2],[−3,1]]

5. Simplify the resulting matrix by distributing the scalar.

A(−1)=[[−2,1],[1.5,−0.5]]

Final Answer

[[1,2],[3,4]](−1)=[[−2,1],[3/2,−1/2]]

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