Find the Determinant [[1-x,1,1],[0,2-x,1],[-4,4,3-x]]

Problem

det(1−x)

Solution

1. Apply cofactor expansion along the first column to take advantage of the zero entry.

2. Calculate the first term by multiplying (1−x) by the determinant of the 2×2 minor matrix.

(1−x)*det(2−x)

3. Calculate the third term by multiplying −4 by the determinant of its 2×2 minor matrix.

−4*det(1)

4. Expand the 2×2 determinants using the formula a*d−b*c

(1−x)*((2−x)*(3−x)−4)−4*(1−(2−x))

5. Simplify the expressions inside the parentheses.

(1−x)*(x2−5*x+6−4)−4*(x−1)

(1−x)*(x2−5*x+2)−4*(x−1)

6. Factor out the common term (1−x) noting that −4*(x−1)=4*(1−x)

(1−x)*(x2−5*x+2+4)

(1−x)*(x2−5*x+6)

7. Factor the quadratic expression x2−5*x+6

(1−x)*(x−2)*(x−3)

8. Distribute the negative sign to write the polynomial in standard form if desired.

−(x−1)*(x−2)*(x−3)

−x3+6*x2−11*x+6

Final Answer

det(1−x)=−x3+6*x2−11*x+6

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