Find the Determinant [[a,-2b,3d],[4a,b,-d],[2a,-b,3d]]

Problem

det(a)

Solution

1. Set up the determinant expansion using the first row.

det(a)=a*det(b)−(−2*b)*det(4*a)+3*d*det(4*a)

2. Calculate the 2×2 determinant for the first term.

a*((b)*(3*d)−(−d)*(−b))=a*(3*b*d−b*d)=a*(2*b*d)=2*a*b*d

3. Calculate the 2×2 determinant for the second term.

2*b*((4*a)*(3*d)−(−d)*(2*a))=2*b*(12*a*d+2*a*d)=2*b*(14*a*d)=28*a*b*d

4. Calculate the 2×2 determinant for the third term.

3*d*((4*a)*(−b)−(b)*(2*a))=3*d*(−4*a*b−2*a*b)=3*d*(−6*a*b)=−18*a*b*d

5. Combine the results of all three terms to find the final value.

2*a*b*d+28*a*b*d−18*a*b*d=12*a*b*d

Final Answer

det(a)=12*a*b*d

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