Concept:Three Basic Elementary Operations of Matrix:Case 1: Interchange of any Two Rows or Two Columns Any 2 columns (or rows) of a matrix can be exchanged. If the ith and jth rows are exchanged, it is shown by Ri ? Rj and if the ith and jth columns are exchanged, it is shown by Ci ? Cj.Case 2: Multiplication of Row or Column by a Non-zero Number The elements of any row (or column) of a matrix can be multiplied by a non-zero number. So if we multiply the ith row of a matrix by a non-zero number k, symbolically it can be denoted by Ri ? kRi. Similarly, for the column, it is given by Ci ? kCi.Case 3: Multiplication of Row or Column by a Non-zero Number and Add the Result to the Other Row or Column The elements of any row (or column) can be added with the corresponding elements of another row (or column) which is multiplied by a non-zero number. So if we add the ith row of a matrix to the jth row which is multiplied by a non-zero number k, symbolically it can be denoted by Ri ? Ri + kRj Similarly, for column it is given by Ci ? Ci + kCjCalculation:Given:(a - b) (b - c) (c - a) = 2 and abc = 6Let ? = \(\begin{vmatrix}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}\)? ? = \(abc\begin{vmatrix}1 & 1 & 1\\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix}\)C1 ? C1 - C2 and C2 ? C2 - C3? ? = \(abc\begin{vmatrix}0 & 0 & 1\\ a-b & b-c & c \\ a^2-b^2 & b^2-c^2 & c^2 \end{vmatrix}\)? ? = \(abc\begin{vmatrix}0 & 0 & 1\\ a-b & b-c & c \\ (a-b)(a+b) & (b-c)(b+c) & c^2 \end{vmatrix}\)? ? = abc (a - b)(b - c) \(\begin{vmatrix}0 & 0 & 1 \\ 1 & 1 & c \\ a+b & b+c & c^2 \end{vmatrix}\)Expanding along a1,3,? ? = abc (a - b)(b - c) [1{(b + c) - (a + b)}]? ? = abc (a - b) (b - c) (c - a)Putting the given values, we get, ? ? = 6 × 2 = 12? The value of \(\begin{vmatrix}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}\) is 12.