= a1(b2c3-b3c2)-b1(a2c3-a3c2)+c1(a2b3-a3b2). In this article, we come across properties of determinants, multiplication of determinants and determinants formula.
as a determinant of order 2 and the four numbers arranged in row and column were called its elements. If we write the coefficients of the equations in the following form
then such an arrangement is called a determinant. In a determinant, horizontal lines are known as rows and vertical lines are known as columns. The shape of every determinant is a square. If a determinant is of order n then it contains n rows and n columns.
\(\begin{array}{l}A=\begin{bmatrix} a &b &c \\ d &e &f \\ g&h &i \end{bmatrix}\end{array} \)
, the value of determinant is = a (ei − fh) − b (di − fg) + c (dh − eg).
Note:
(i) The number of elements in a determinant of order n is n2.
(ii) A determinant of order 1 is the number itself.
Properties of Determinants
There will be no change in the value of the determinant if the rows and columns are interchanged.
Suppose any two rows or columns of a determinant are interchanged, then its sign changes.
If any two rows or columns of a determinant are the same, then the determinant is 0.
If any row or column of the determinant is multiplied by a variable k, then its value is multiplied by k.
Say if some or all elements of a row or column are expressed as the sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.
A determinant is called a Symmetric Determinant if
\(\begin{array}{l}{{a}_{i\,j}}={{a}_{j\,i}},\forall \,i,j \;e.g.\; [\left| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \\ \end{matrix} \right|\end{array} \)
Skew Symmetric Determinant
A determinant is called a skew symmetric determinant if
(i) The two determinants to be multiplied must be of the same order.
(ii) To get the
\(\begin{array}{l}{{T}_{mn}}\end{array} \)
(term in the mth row nth column) in the product, Take the mth row of the 1st determinant and multiply it by the corresponding terms of the nth column of the 2nd determinant and add.
(iii) This method is the row by column multiplication rule for the product of 2 determinants of the nth order determinant.
(iv) IfΔ′ is the determinant formed by replacing the elements of a Δ of order n by their corresponding co-factors then
Illustration 3: Reduce the power of the determinant
\(\begin{array}{l}{{\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|}^{2}}\end{array} \)
to 1.
Solution:
By multiplying the given determinant two times we get the determinant as required.
\(\begin{array}{l}{{\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|}^{2}}=\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|\left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \\ \end{matrix} \right|\Rightarrow \left| \begin{matrix} {{b}^{2}}+{{c}^{2}} & ab & ac \\ ba & {{c}^{2}}+{{a}^{2}} & bc \\ ca & cb & {{a}^{2}}+{{b}^{2}} \\ \end{matrix} \right|\end{array} \)
