Find the Determinant [[a,b,c],[d,e,f],[5g,5h,5i]]

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 factored out of the determinant.

3. Factor out the constant 5 from the third row.

5⋅det(a)

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

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

5. Multiply the expanded expression by the factor of 5

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

Final Answer

det(a)=5*a*e*i−5*a*ƒ*h−5*b*d(i)+5*b*ƒ*g+5*c*d(h)−5*c*e*g

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