Find the Determinant A=[[x,3,x^2],[-3,5x,0],[4,x^3,1]]

Problem

det(x)

Solution

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

A=[[x,3,x2],[−3,5*x,0],[4,x3,1]]

2. Apply the cofactor expansion along the first row.

det(A)=x*|[5*x,0],[x3,1]|−3*|[−3,0],[4,1]|+x2*|[−3,5*x],[4,x3]|

3. Calculate the 2×2 determinants for each term.

det(A)=x*(5*x(1)−0*(x3))−3*(−3*(1)−0*(4))+x2*(−3*(x3)−5*x(4))

4. Simplify the expressions inside the parentheses.

det(A)=x*(5*x)−3*(−3)+x2*(−3*x3−20*x)

5. Distribute the terms outside the parentheses.

det(A)=5*x2+9−3*x5−20*x3

6. Rearrange the polynomial in descending order of powers.

det(A)=−3*x5−20*x3+5*x2+9

Final Answer

det(x)=−3*x5−20*x3+5*x2+9

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