Find the Determinant of the Resulting Matrix

Problem

det([[10,5,18],[4,−15,−5],[1,8,0]]*[[−7,2,−14],[17,3,−13],[16,15,10]])

Solution

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

2. Calculate the determinant of the first matrix A using expansion by minors along the third row.

det(A)=1⋅det(5)−8⋅det(10)+0

det(A)=1*(5*(−5)−18*(−15))−8*(10*(−5)−18*(4))

det(A)=1*(−25+270)−8*(−50−72)

det(A)=245−8*(−122)

det(A)=245+976=1221

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

det(B)=−7⋅det(3)−2⋅det(17)+(−14)⋅det(17)

det(B)=−7*(30−(−195))−2*(170−(−208))−14*(255−48)

det(B)=−7*(225)−2*(378)−14*(207)

det(B)=−1575−756−2898=−5229

4. Multiply the two determinants to find the determinant of the product matrix.

det(A*B)=1221⋅(−5229)

det(A*B)=−6384609

Final Answer

det([[10,5,18],[4,−15,−5],[1,8,0]]*[[−7,2,−14],[17,3,−13],[16,15,10]])=−6384609

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