Concept:Expansion of determinant \(\begin{vmatrix}a_{11} & b_{12} & c_{13} \\ a_{21} & b_{22} & c_{23} \\ a_{31} & b_{32} & c_{33} \end{vmatrix}\) is given by ? = a11 a22 a33 - a11 a23 a32 - a12 a21 a33 + a12 a23 a31 + a13 a21 a32 - a13 a31 a22Three Basic Elementary Operations of MatrixCase 1: Interchange of any Two Rows or Two ColumnsAny 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 NumberThe 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 ColumnThe 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 + kCj. Formula used:a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - caIf (a + b + c) = 0 then a3 + b3 + c3 = 3abc Calculation:Let A = \(\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}\)R1 ? R1 + R2 + R3? A = \(\begin{vmatrix}a+b+c & a+b+c & a+b+c \\ b & c & a \\ c & a & b\end{vmatrix}\)C2 ? C2 - C1 and C3 ? C3 - C1? A = (a + b + c) \(\begin{vmatrix}1 & 0 & 0 \\ b & c-b & a-b \\ c & a-c & b-c\end{vmatrix}\)Expanding along a1,1? A = (a + b + c)[1{(c - b)(b - c) - (a - b)(a - c)}]? A = (a + b + c)[bc - c2 - b2 + bc - (a2 - ac - ab + bc)]? A = (a + b + c)[- c2 - b2 - a2 + ac + ab + bc]? A = - (a + b + c)[a2 + b2 + a2 - ab - bc - ac]? A value can be zero if (a + b + c) or (a2 + b2 + a2 - ab - bc - ac) is 0.We know that,a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca? a3 + b3 + c3 = 3abc ? 1, 2 and 3 are correct.