# JEE Main & Advanced Mathematics Determinants & Matrices Some Special Determinants

Some Special Determinants

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.,  $\left| \,\begin{matrix} 0 & 3 & -1 \\ -3 & 0 & 5 \\ 1 & -5 & 0 \\ \end{matrix}\, \right|$

• Every diagonal element of a skew symmetric determinant is always zero.

• The value of a skew symmetric determinant of even order is always a perfect square and that of odd order is always zero.

(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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