By using properties of determinants, in Exercises 8 to 14 ,...

By using properties of determinants, in Exercises 8 to 14 , show that: (i) $\left|\begin{array}{lll}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2}\end{array}\right|=(a-b)(b-c)(c-a)$ (ii) $\left|\begin{array}{lll}1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3}\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)$

By using properties of determinants, in Exercises 8 to 14 , show that:
(i) $\left|\begin{array}{lll}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2}\end{array}\right|=(a-b)(b-c)(c-a)$
(ii) $\left|\begin{array}{lll}1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3}\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)$

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Key Concepts

Determinants

Determinants are scalar values computed from a square matrix and are used to determine properties such as the invertibility of the matrix, areas, and volumes. They play a central role in linear algebra, particularly in solving systems of linear equations and understanding matrix transformations.

Vandermonde Determinant

A Vandermonde determinant is a special type of determinant that arises from matrices whose rows (or columns) contain successive powers of variables. Its evaluation leads to a product of pairwise differences among the variables, showcasing the inherent symmetry and factorization properties of polynomials.

Properties of Determinants

Properties such as linearity, the effect of row and column operations, and the impact of interchanging rows or columns are fundamental in simplifying determinant evaluations and in proving identities. These properties allow for common factors to be factored out and for expressions to be restructured in a more manageable form.

Row Operations

Row operations, including adding a multiple of one row to another and swapping rows, are instrumental tools for transforming a determinant into a simpler or more revealing form. They preserve or predictably change the determinant in known ways, enabling the reduction of complex determinants into products of simpler expressions.

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