Find the Determinant of the Resulting Matrix

Problem

det([[6,2,3],[4,5,6],[2,3,4]]+[[1,3,5],[2,4,6],[5,10,15]])

Solution

1. Add the matrices by summing the corresponding entries of each matrix.

[[6,2,3],[4,5,6],[2,3,4]]+[[1,3,5],[2,4,6],[5,10,15]]=[[7,5,8],[6,9,12],[7,13,19]]

2. Apply the determinant formula for a 3×3 matrix using cofactor expansion along the first row.

det(A)=(a_11)*((a_22)*(a_33)−(a_23)*(a_32))−(a_12)*((a_21)*(a_33)−(a_23)*(a_31))+(a_13)*((a_21)*(a_32)−(a_22)*(a_31))

3. Substitute the values from the resulting matrix into the expansion formula.

det(=)*7*(9⋅19−12⋅13)−5*(6⋅19−12⋅7)+8*(6⋅13−9⋅7)

4. Calculate the products inside the parentheses.

det(=)*7*(171−156)−5*(114−84)+8*(78−63)

5. Simplify the expressions within the parentheses.

det(=)*7*(15)−5*(30)+8*(15)

6. Perform the final arithmetic to find the determinant value.

det(=)*105−150+120

det(=)*75

Final Answer

det([[6,2,3],[4,5,6],[2,3,4]]+[[1,3,5],[2,4,6],[5,10,15]])=75

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