If α, β, γare the roots of x^3+p x+q=0, then

Key Concepts

Determinants and Linear Dependence

The determinant of a matrix is a scalar value that reflects properties of linear transformation, such as invertibility and linear independence of rows or columns. When determinants are zero, it implies that there is linear dependence among the rows or columns, often arising from underlying symmetrical relations among the entries. This principle is crucial in recognizing and proving certain structured outcomes in matrices constructed from polynomial roots.

Vieta’s Formulas

Vieta’s formulas provide a connection between the coefficients of a polynomial and the sums and products of its roots. For a cubic polynomial with no x² term, these formulas imply that the sum of the roots is zero. This relation is essential in determining the behavior of systems created from the roots, such as matrices, and simplifies computations by introducing symmetry.

Circulant Matrices

Circulant matrices are a subset of matrices where each row is a cyclic shift of the previous one. The structure of these matrices allows for determinant evaluation using properties of roots of unity, leading to elegant and compact results. The inherent symmetry of circulant matrices often results in cancellations or zero values when combined with conditions from related polynomial equations.

Transcript

00:01 So in the given question we are told that alpha, beta and gamma are the roots of the root of the cubic equation x cubed plus px plus q equal to 0 and now we are given a determinant as alpha beta gamma beta gamma and gamma alpha alpha and gamma alpha beta alpha beta we are told to evaluate this determinant, right? so first we can write some relations in between alpha, beta and gamma since they are the roots of the cubic equation, right? so if a, b, c are the roots of a cubic equation, we can write, we can take pqr, right? so pqr are the roots of an equation, cubic equation, a x cubed plus b, square.

01:03 Plus cx plus d equal to 0, then we can write a few relations in between the roots p, q and r, right? so the relations are p plus q plus r is equal to minus b by a and we can write pq plus qr plus pr as equal to c by a and we can all.

01:36 Also write p q r as equal to the negative of the constant divided by a.

01:44 So this is a standard result or we can say a property of the roots of a cubic equation of this form, right? so over here we have the equation x cubed plus 0 x squared plus px plus q plus q equal to 0 right and the roots are alpha beta and gamma so the relations that we can form in between alpha beta and gamma are alpha plus beta plus gamma is equal to minus b which is 0 over here so alpha plus beta plus gamma is equal to 0 next we have alpha beta plus beta plus gamma which would be equal to c by a where c is the coefficient of x over here.

02:38 So the coefficient of x over here is p divided by a where a is the coefficient of x cube over here over here in this equation we have the coefficient of x cube as one.

02:50 So alpha beta plus beta gamma plus alpha gamma is equal to p...

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