Find the Determinant

Problem

det(a2)

Solution

1. Perform column operations to simplify the matrix by subtracting the first column from the second and third columns ((C_2)→(C_2)−(C_1) and (C_3)→(C_3)−(C_1).

det(a2)

2. Simplify the entries in the second and third columns using factoring and basic arithmetic.

det(a2)

3. Factor out common terms (b−a) from the second column and (c−a) from the third column.

(b−a)*(c−a)*det(a2)

4. Perform another column operation by subtracting the second column from the third ((C_3)→(C_3)−(C_2).

(b−a)*(c−a)*det(a2)

5. Factor out (c−b) from the third column.

(b−a)*(c−a)*(c−b)*det(a2)

6. Expand the determinant along the third column using minors.

(b−a)*(c−a)*(c−b)*(1⋅(2*b*c−(−c)*(a−b−c))−1⋅(2*a2−(b+a)*(a−b−c))+0)

7. Simplify the polynomial inside the brackets.

2*b*c+a*c−b*c+c2−2*a2+(a*b+a*c+b2−b*c+a2−a*b−a*c)

b*c+a*c+c2−2*a2+b2−b*c+a2

a*c+c2+b2−a2

8. Combine the factors and adjust signs to reach the standard cyclic form (a−b)*(b−c)*(c−a)

−(a−b)*(b−c)*(c−a)*(a2−b2−c2−a*c)

Final Answer

det(a2)=(a−b)*(b−c)*(c−a)*(a+b+c)2

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