Simplify the Matrix [[1,a+1,a^2],[1-a,1-2a,0],[1,1+a,a]]

Problem

[[1,a+1,a2],[1−a,1−2*a,0],[1,1+a,a]]

Solution

1. Identify the matrix and the goal to simplify it, which typically involves finding its determinant or performing row reductions if it were part of a system. Since the task is to "simplify" the matrix expression, we will calculate its determinant to provide a scalar representation of the matrix.

2. Apply the cofactor expansion along the third column because it contains a zero, which simplifies the calculation.

3. Calculate the first term of the expansion using the element a2 at position (1,3)

a2*det(1−a)

4. Evaluate the 2×2 determinant for the first term.

(1−a)*(1+a)−(1)*(1−2*a)

1−a2−1+2*a

2*a−a2

5. Calculate the second term of the expansion using the element a at position (3,3) Note that the sign for position (2,3) is negative, but the element is 0

a*det(1)

6. Evaluate the 2×2 determinant for the second term.

(1)*(1−2*a)−(1−a)*(a+1)

1−2*a−(1−a2)

a2−2*a

7. Combine the terms to find the total determinant value.

a2*(2*a−a2)+a*(a2−2*a)

2*a3−a4+a3−2*a2

−a4+3*a3−2*a2

8. Factor the resulting polynomial to reach the simplest form.

−a2*(a2−3*a+2)

−a2*(a−1)*(a−2)

Final Answer

det(1)=−a2*(a−1)*(a−2)

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