Find the Determinant [[4-x,-2,0],[-3,7-x,0],[5,7,3-x]]

Problem

det(4−x)

Solution

1. Identify the best row or column for expansion. The third column contains two zeros, making it the most efficient choice for cofactor expansion.

2. Apply the cofactor expansion along the third column. The determinant is the product of the element 3−x and its corresponding minor.

D=(3−x)⋅det(4−x)

3. Calculate the determinant of the 2×2 matrix using the formula a*d−b*c

det(4−x)=(4−x)*(7−x)−(−2)*(−3)

4. Expand the algebraic expression inside the brackets.

(4−x)*(7−x)−6=28−4*x−7*x+x2−6

5. Simplify the quadratic expression by combining like terms.

x2−11*x+22

6. Multiply the simplified quadratic by the remaining factor (3−x) from the first step.

D=(3−x)*(x2−11*x+22)

7. Distribute the terms to find the final polynomial.

D=3*x2−33*x+66−x3+11*x2−22*x

8. Combine like terms and write the polynomial in standard form.

D=−x3+14*x2−55*x+66

Final Answer

det(4−x)=−x3+14*x2−55*x+66

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