DETERMINANT
1. MINORS :
The minor of a given element of determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands. For example, the minor of \(a _{1}\) in \(\left|\begin{array}{lll} a _{1} & b _{1} & c _{1} \\ a _{2} & b _{2} & c _{2} \\ a _{3} & b _{3} & c _{3}\end{array}\right|\) is \(\left|\begin{array}{ll} b _{2} & c _{2} \\ b _{3} & c _{3}\end{array}\right| \&\) the minor of \(b_{2}\) is \(\quad\left|\begin{array}{ll}a_{1} & c_{1} \\ a_{3} & c_{3}\end{array}\right|\).
Hence a determinant of order three will have " 9 minors".
2. COFACTORS :
If \(M _{ ij }\) represents the minor of the element belonging to \(i ^{\text {th }}\) row and \(j ^{\text {th }}\) column then the cofactor of that element : \(C_{i j}=(-1)^{i+j} \cdot M_{i j}\) Important Note :
Consider \(\Delta=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3}\end{array}\right|\)
Let \(A_{1}\) be cofactor of \(a_{1}, B_{2}\) be cofactor of \(b_{2}\) and so on, then,
(i) \(a _{1} A _{1}+ b _{1} B _{1}+ c _{1} C _{1}\)\(= a _{1} A _{1}+ a _{2} A _{2}+ a _{3} A _{3}=\)\(\ldots \ldots \ldots \ldots . .=\Delta\)
(ii) \(a _{2} A _{1}+ b _{2} B _{1}+ c _{2} C _{1}\)\(= b _{1} A _{1}+ b _{2} A _{2}+ b _{3} A _{3}\)\(=\ldots \ldots \ldots \ldots .=0\)
3. PROPERTIES OF DETERMINANTS:
(a) The value of a determinants remains unaltered, if the rows & corresponding columns are interchanged.
(b) If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g.
Let \(\quad D=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\, \& \, D^{\prime}=\left|\begin{array}{lll}a_{2} & b_{2} & c_{2} \\ a_{1} & b_{1} & c_{1} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\) Then \(D^{\prime}=-D\).
(c) If a determinant has any two rows (or columns) identical or in same proportion, then its value is zero.
(d) If all the elements of any row (or column) be multiplied by the same number, then the determinant is multiplied by that number.
(e) \(\left|\begin{array}{ccc} a _{1}+ x & b _{1}+ y & c _{1}+ z \\a _{2} & b _{2} & c _{2} \\a _{3} & b _{3} & c _{3}\end{array}\right|\)\(=\left|\begin{array}{ccc} a _{1} & b _{1} & c _{1} \\a _{2} & b _{2} & c _{2} \\a _{3} & b _{3} & c _{3}\end{array}\right|+\left|\begin{array}{ccc} x & y & z \\a _{2} & b _{2} & c _{2} \\a _{3} & b _{3} & c _{3}\end{array}\right|\)
(f) The value of a determinant is not altered by adding to the elements of any row (or column ) the same multiples of the corresponding elements of any other row (or column) e.g.
\(\begin{array}{ll}\text { Let } & D =\left|\begin{array}{lll} a _{1} & b _{1} & c _{1} \\a _{2} & b _{2} & c _{2} \\a _{3} & b _{3} & c _{3}\end{array}\right| \\D ^{\prime} & =\left|\begin{array}{} a _{1}+ ma _{2} & b _{1}+ mb _{2} & c _{1}+ mc _{2} \\a _{2} & b _{2} & c _{2} \\a _{3}+ na _{1} & b _{3}+ nb _{1} & c _{3}+ nc _{1}\end{array}\right| .\end{array}\) \(\text { Then } D ^{\prime}= D .\)
Note :- While applying this property ATLEAST ONE ROW (OR COLUMN) must remain unchanged.
(g) If the elements of a determinant \(\Delta\) are rational function of \(x\) and two rows (or columns) become identical when \(x = a\), then \(x\) \(-a\) is a factor of \(\Delta\).
Again, if \(r\) rows become identical when a is substituted for \(x\), then \((x-a)^{r-1}\) is a factor of \(\Delta\).
(h) If \(D ( x )=\left|\begin{array}{ccc} f _{1} & f _{2} & f _{3} \\ g _{1} & g _{2} & g _{3} \\ h _{1} & h _{2} & h _{3}\end{array}\right|\), where \(f _{ r }, g _{ r }, h _{ r } ; r =1,2,3\) are three differentiable functions.
\(\text { then } \dfrac{ d }{ dx } D ( x )\)\(=\left|\begin{array}{lll} f _{1}^{\prime} & f _{2}^{\prime} & f _{3}^{\prime} \\g _{1} & g _{2} & g _{3} \\h _{1} & h _{2} & h _{3}\end{array}\right|+\left|\begin{array}{ccc} f _{1} & f _{2} & f _{3} \\g _{1}^{\prime} & g ^{\prime}{ }_{2} & g _{3}^{\prime} \\h _{1} & h _{2} & h _{3}\end{array}\right|+\left|\begin{array}{ccc} f _{1} & f _{2} & f _{3} \\g _{1} & g _{2} & g _{3} \\h _{1}^{\prime} & h _{2}^{\prime} & h _{3}^{\prime}\end{array}\right|\)
4. MULTIPLICATION OF TWO DETERMINANTS :
\(\left|\begin{array}{ll} a _{1} & b _{1} \\ a _{2} & b _{2}\end{array}\right| \times\left|\begin{array}{cc}l_{1} & m _{1} \\ l_{2} & m _{2}\end{array}\right|\)\(=\left|\begin{array}{ll} a _{1} l_{1}+ b _{1} l_{2} & a _{1} m _{1}+ b _{1} m _{2} \\ a _{2} l_{1}+ b _{2} l_{2} & a _{2} m _{1}+ b _{2} m _{2}\end{array}\right|\)
Similarly two determinants of order three are multiplied.
(a) Here we have multiplied row by column. We can also multiply row by row, column by row and column by column.
(b) If D' is the determinant formed by replacing the elements of determinant D of order \(n\) by their corresponding cofactors then \(D ^{\prime}= D ^{ n -1}\)
5. SPECIAL DETERMINANTS :
(a) Symmetric Determinant:
Elements of a determinant are such that \(a_{i j}=a_{j i}\)
\(\text { e.g. }\left|\begin{array}{lll}a & h & g \\h & b & f \\g & f & c\end{array}\right|=a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}\)
(b) Skew Symmetric Determinant :
If \(a_{i j}=-a_{j i}\) then the determinant is said to be a skew symmetric determinant. Here all the principal diagonal elements are zero. The value of a skew symmetric determinant of odd order is zero and of even order is perfect square.
\(\text { e.g. }\left|\begin{array}{llr}0 & b & -c \\-b & 0 & a \\c & -a & 0\end{array}\right|=0\)
(c) Other Important Determinants:
(i) \(\left|\begin{array}{lll}1 & 1 & 1 \\ a & b & c \\ b c & a c & a b\end{array}\right|=\left|\begin{array}{lll}1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2}\end{array}\right|=(a-b)(b-c)(c-a)\)
(ii) \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=-\left(a^{3}+b^{3}+c^{3}-3 a b c\right)\)
6. SYSTEM OF EQUATION :
(a) System of equation involving two variable :
If \(\Delta_{1}=\left|\begin{array}{ll}b_{1} & c_{1} \\ b_{2} & c_{2}\end{array}\right|, \Delta_{2}=\left|\begin{array}{ll}c_{1} & a_{1} \\ c_{2} & a_{2}\end{array}\right|, \Delta=\left|\begin{array}{ll}a_{1} & b_{1} \\ a_{2} & b_{2}\end{array}\right|\), then \(x=\frac{\Delta_{1}}{\Delta}, y=\frac{\Delta_{2}}{\Delta}\)
(b) System of equations involving three variables:
\(a_{1} x+b_{1} y+c_{1} z=d_{1}\)
\(a_{2} x+b_{2} y+c_{2} z=d_{2} \)
\(a_{3} x+b_{3} y+c_{3} z=d_{3}\)
To solve this system we first define following determinants
\(\Delta=\left|\begin{array}{lll} a _{1} & b _{1} & c _{1} \\a _{2} & b _{2} & c _{2} \\a _{3} & b _{3} & c _{3}\end{array}\right|, \quad \Delta_{1}=\left|\begin{array}{lll} d _{1} & b _{1} & c _{1} \\d _{2} & b _{2} & c _{2} \\d _{3} & b _{3} & c _{3}\end{array}\right|\)
\(\Delta_{2}=\left|\begin{array}{lll} a _{1} & d _{1} & c_{1} \\a _{2} & d _{2} & c _{2} \\a _{3} & d _{3} & c _{3}\end{array}\right|, \quad \Delta_{3}=\left|\begin{array}{lll} a _{1} & b_{1} & d _{1} \\a _{2} & b _{2} & d _{2} \\a _{3} & b _{3} & d _{3}\end{array}\right|\)
Note:
(i) Trivial solution : In the solution set of system of equation if all the variables assumes zero, then such a solution set is called Trivial solution otherwise the solution is called non-trivial solution.
(ii) If \(d _{1}= d _{2}= d _{3}=0\) then system of linear equation is known as system of Homogeneous linear equation which always posses atleast one solution \((0,0,0)\).
(iii) If system of homogeneous linear equation posses non-zero/nontrivial solution then \(\Delta=0\).
In such case given system has infinite solutions.
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