JEE Main & Advanced Mathematics Determinants & Matrices Some Special Determinants

(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)  \[\left| \,\begin{matrix} a & b & c  \\ b & c & a  \\ c & a & b  \\ \end{matrix}\, \right|=-({{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc)\]