Given that a² + b² + c² = -2, define f(x) as the determinant of the 3x3 matrix with diagonal entries (1 + a²*x), (1 + b²*x), (1 + c²*x) and off-diagonal entries: row 1 has (1+b²)*x and (1+c²)*x; row 2 has (1+a²)*x and (1+c²)*x; row 3 has (1+a²)*x and (1+b²)*x. What is the degree of the pol

Question

Given that a² + b² + c² = -2, define f(x) as the determinant of the 3x3 matrix with diagonal entries (1 + a²*x), (1 + b²*x), (1 + c²*x) and off-diagonal entries: row 1 has (1+b²)*x and (1+c²)*x; row 2 has (1+a²)*x and (1+c²)*x; row 3 has (1+a²)*x and (1+b²)*x. What is the degree of the polynomial f(x)?

Accepted Answer

2. The matrix can be written as (1-x)*I + x*(J + P) where J is the all-ones matrix and P accounts for a²,b²,c² terms. Expanding the determinant, the cubic term coefficient involves det of a rank-1 update and vanishes due to a²+b²+c²=-2. The resulting polynomial has degree 2.

Solution

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