Find the Determinant of the Resulting Matrix

Problem

det([[−7,−39,6],[7,14,22],[19,11,82]]+[[8,5,36],[24,54,72],[24,62,56]])

Solution

1. Add the two matrices by summing their corresponding entries to find the resulting matrix A

A=[[−7+8,−39+5,6+36],[7+24,14+54,22+72],[19+24,11+62,82+56]]

A=[[1,−34,42],[31,68,94],[43,73,138]]

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

det(A)=1*|[68,94],[73,138]|−(−34)*|[31,94],[43,138]|+42*|[31,68],[43,73]|

3. Calculate the 2×2 determinants using the formula a*d−b*c

|[68,94],[73,138]|=(68)*(138)−(94)*(73)=9384−6862=2522

|[31,94],[43,138]|=(31)*(138)−(94)*(43)=4278−4042=236

|[31,68],[43,73]|=(31)*(73)−(68)*(43)=2263−2924=−661

4. Substitute these values back into the expansion and simplify.

det(A)=1*(2522)+34*(236)+42*(−661)

det(A)=2522+8024−27762

det(A)=10546−27762=−17216

Final Answer

det([[−7,−39,6],[7,14,22],[19,11,82]]+[[8,5,36],[24,54,72],[24,62,56]])=−17216

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