Find the Determinant of the Resulting Matrix

Problem

det([[34,23,49],[15,3,24],[19,20,25]]−3*[[8,7,11],[3,6,3],[5,1,8]]+[[2,1,3],[1,−1,2],[1,2,1]])

Solution

1. Multiply the second matrix by the scalar 3

3*[[8,7,11],[3,6,3],[5,1,8]]=[[24,21,33],[9,18,9],[15,3,24]]

2. Subtract the result from the first matrix.

[[34,23,49],[15,3,24],[19,20,25]]−[[24,21,33],[9,18,9],[15,3,24]]=[[10,2,16],[6,−15,15],[4,17,1]]

3. Add the third matrix to the current result to find the final matrix M

M=[[10,2,16],[6,−15,15],[4,17,1]]+[[2,1,3],[1,−1,2],[1,2,1]]=[[12,3,19],[7,−16,17],[5,19,2]]

4. Calculate the determinant of M using expansion by the first row.

det(M)=12*((−16)*(2)−(17)*(19))−3*((7)*(2)−(17)*(5))+19*((7)*(19)−(−16)*(5))

5. Simplify the arithmetic within the expansion.

det(M)=12*(−32−323)−3*(14−85)+19*(133+80)

det(M)=12*(−355)−3*(−71)+19*(213)

det(M)=−4260+213+4047

det(M)=0

Final Answer

det([[34,23,49],[15,3,24],[19,20,25]]−3*[[8,7,11],[3,6,3],[5,1,8]]+[[2,1,3],[1,−1,2],[1,2,1]])=0

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