Find the Determinant [[-3i,2j,4k],[5i,1j,6k],[2i,3j,1k]]

Problem

det(−3*i)

Solution

1. Set up the determinant expansion along the first row using the cofactor method.

2. Calculate the first minor by multiplying −3*i by the determinant of the 2×2 matrix remaining after removing the first row and first column.

(−3*i)⋅((1*j)*(1*k)−(6*k)*(3*j))=−3*i*(j*k−18*k*j)

3. Calculate the second minor by subtracting 2*j multiplied by the determinant of the 2×2 matrix remaining after removing the first row and second column.

−(2*j)⋅((5*i)*(1*k)−(6*k)*(2*i))=−2*j*(5*i*k−12*k*i)

4. Calculate the third minor by adding 4*k multiplied by the determinant of the 2×2 matrix remaining after removing the first row and third column.

(4*k)⋅((5*i)*(3*j)−(1*j)*(2*i))=4*k*(15*i*j−2*j*i)

5. Simplify the terms by combining the products of the unit vectors i,j,k

−3*i*(j*k−18*k*j)−2*j*(5*i*k−12*k*i)+4*k*(15*i*j−2*j*i)

6. Distribute the coefficients into the parentheses.

−3*i*j*k+54*i*k*j−10*j*i*k+24*j*k*i+60*k*i*j−8*k*j*i

7. Apply the properties of the Levi-Civita symbol or cross product rules for unit vectors (i*j*k=j*k*i=k*i*j=1 and i*k*j=k*j*i=j*i*k=−1.

−3*(1)+54*(−1)−10*(−1)+24*(1)+60*(1)−8*(−1)

8. Sum the resulting numerical values.

−3−54+10+24+60+8=45

Final Answer

det(−3*i)=45

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