Find the Determinant of the Resulting Matrix

Problem

det([[1,0,−3],[−2,1,5],[3,4,2]]*[[2,3,1],[3,−4,5],[1,−1,−2]])

Solution

1. Apply the property of determinants for the product of two matrices, which states that det(A*B)=det(A)*det(B)

2. Calculate the determinant of the first matrix A using cofactor expansion along the first row.

det(A)=1*(1*(2)−5*(4))−0*(−2*(2)−5*(3))+(−3)*(−2*(4)−1*(3))

det(A)=1*(2−20)−3*(−8−3)

det(A)=−18+33=15

3. Calculate the determinant of the second matrix B using cofactor expansion along the first row.

det(B)=2*((−4)*(−2)−5*(−1))−3*(3*(−2)−5*(1))+1*(3*(−1)−(−4)*(1))

det(B)=2*(8+5)−3*(−6−5)+1*(−3+4)

det(B)=2*(13)−3*(−11)+1*(1)

det(B)=26+33+1=60

4. Multiply the determinants of the individual matrices to find the determinant of the product.

det(A*B)=15×60=900

Final Answer

det([[1,0,−3],[−2,1,5],[3,4,2]]*[[2,3,1],[3,−4,5],[1,−1,−2]])=900

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