Symmetric & Skew Symmetric Determinants

Symmetric determinant :

If the elements of a determinant are such that aij = aji ,

then the determinant is said to be a Symmetric determinant.

The elements situated at equal distances from the diagonal are equal both in magnitude and sign. e.g.

$ \large \Delta (x) = \left| \begin{array}{ccc} a & h & g \\ h & b & r \\ g & f & c  \end{array} \right| = abc + 2fgh – af^2 – bg^2 – ch^2 $

(Remember this formula for quick calculations)

Skew Symmetric determinant:

If aij = −aji then the determinant is said to be a Skew symmetric determinant. That is all the diagonal elements are zero and the elements situated at equal distances from the diagonal are equal in magnitude but opposite in sign. The value of a skew symmetric determinant of odd order is zero. e.g.

$ \large  \left| \begin{array}{ccc} 0 & b & -c \\ -b & 0 & a \\ c & -a & 0  \end{array} \right| = 0 $

Circulant Determinants:

The elements of the rows (or columns) are in cyclic arrangement.

$ \large  \left| \begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b  \end{array} \right| = -(a^3 + b^3 + c^3 – 3abc)$

Other Important Determinants :

$ \large \Delta  = \left| \begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ bc & ac & ab  \end{array} \right| =  \left| \begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2  \end{array} \right| = (a-b)(b-c)(c-a)$

$ \large \Delta  = \left| \begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3  \end{array} \right|  = (a-b)(b-c)(c-a)(a+b+c) $

$ \large \Delta  = \left| \begin{array}{ccc} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3  \end{array} \right|  = (a-b)(b-c)(c-a)(ab+bc+ca) $

$ \large \Delta  = \left| \begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^4 & b^4 & c^4 \end{array} \right|  = (a-b)(b-c)(c-a)(a^2+b^2+c^2-ab-bc – ca) $

Illustration : Prove that :

$ \large \Delta  = \left| \begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ bc+a^2 & ac+b^2 & ab+c^2 \end{array} \right|  = 2(a-b)(b-c)(c-a) $

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