Category : JEE Main & Advanced
(1) Symmetric determinant
A determinant is called symmetric determinant if for its every element \[{{a}_{ij}}\,=\,\,\,{{a}_{ji\,}}\forall \,\,i,\,j\] e.g., \[\left| \,\begin{matrix} a & h & g \\ h & b & f \\ g & f & c \\ \end{matrix}\, \right|\].
(2) Skew-symmetric determinant : A determinant is called skew symmetric determinant if for its every element \[{{a}_{ij}}\,=\,-\,{{a}_{ji\,\,}}\forall \,i,\,j\] e.g., \[\left| \,\begin{matrix} 0 & 3 & -1 \\ -3 & 0 & 5 \\ 1 & -5 & 0 \\ \end{matrix}\, \right|\]
(3) Cyclic order : If elements of the rows (or columns) are in cyclic order. i.e., (i) \[\left| \,\begin{matrix} 1 & a & {{a}^{2}} \\ 1 & b & {{b}^{2}} \\ 1 & c & {{c}^{2}} \\ \end{matrix}\, \right|=(a-b)(b-c)(c-a)\]
(ii) \[\left| \,\begin{matrix} a & b & c \\ {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ bc & ca & ab \\ \end{matrix}\, \right|=\left| \,\begin{matrix} 1 & 1 & 1 \\ {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ {{a}^{3}} & {{b}^{3}} & {{c}^{3}} \\ \end{matrix}\, \right|\]
\[=(a-b)(b-c)(c-a)(ab+bc+ca)\]
(iii) \[\left| \,\begin{matrix} a & bc & abc \\ b & ca & abc \\ c & ab & abc \\ \end{matrix}\, \right|=\left| \,\begin{matrix} a & {{a}^{2}} & {{a}^{3}} \\ b & {{b}^{2}} & {{b}^{3}} \\ c & {{c}^{2}} & {{c}^{3}} \\ \end{matrix}\, \right|=abc(a-b)(b-c)(c-a)\]
(iv) \[\left| \,\begin{matrix} 1 & 1 & 1 \\ a & b & c \\ {{a}^{3}} & {{b}^{3}} & {{c}^{3}} \\ \end{matrix}\, \right|=(a-b)(b-c)(c-a)(a+b+c)\]
(v) \[\left| \,\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix}\, \right|=-({{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc)\]
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