Find the Determinant of the Resulting Matrix

Problem

det([[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]−[[1,0,0],[0,1,0],[0,0,1]])

Solution

1. Subtract the identity matrix from the first matrix by subtracting 1 from each diagonal element.

A=[[0.3−1,0.2,0.3],[0.4,0.3−1,0.3],[0.3,0.3,0.2−1]]

A=[[−0.7,0.2,0.3],[0.4,−0.7,0.3],[0.3,0.3,−0.8]]

2. Apply the cofactor expansion formula along the first row to find the determinant.

det(A)=−0.7*det(−0.7)−0.2*det(0.4)+0.3*det(0.4)

3. Calculate the 2×2 determinants for each term.

det(−0.7)=(−0.7)*(−0.8)−(0.3)*(0.3)=0.56−0.09=0.47

det(0.4)=(0.4)*(−0.8)−(0.3)*(0.3)=−0.32−0.09=−0.41

det(0.4)=(0.4)*(0.3)−(−0.7)*(0.3)=0.12+0.21=0.33

4. Combine the results to find the final determinant value.

det(A)=−0.7*(0.47)−0.2*(−0.41)+0.3*(0.33)

det(A)=−0.329+0.082+0.099

det(A)=−0.148

Final Answer

det([[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]−[[1,0,0],[0,1,0],[0,0,1]])=−0.148

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