Find the Determinant [[a,b,c],[d,e,f],[4g,4h,4i]]

Problem

det(a)

Solution

1. Identify the matrix and the scalar factor in the third row.

2. Apply the property of determinants that allows a common factor from a single row to be moved outside the determinant.

3. Factor out the constant 4 from the third row.

det(a)=4⋅det(a)

4. Expand the remaining 3×3 determinant using the standard formula for a determinant.

det(a)=a*(e*i−ƒ*h)−b*(d(i)−ƒ*g)+c*(d(h)−e*g)

5. Distribute the scalar 4 across the expanded expression.

4*(a*e*i−a*ƒ*h−b*d(i)+b*ƒ*g+c*d(h)−c*e*g)

Final Answer

det(a)=4*a*e*i−4*a*ƒ*h−4*b*d(i)+4*b*ƒ*g+4*c*d(h)−4*c*e*g

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