If the equations a_{1}x + b_{1} = 0 and a_{2}x + b_{2} = 0 are satisfied by the same value of x, then a_{1}b_{2} – a_{2}b_{1} = 0.
The expression a_{1}b_{2} – a_{2}b_{1} is called a determinant of the second order, and is denoted by:
A determinant of the second order consists of two rows and two columns.
Consider the system of equations:
If these equations are satisfied by the same values of x and y, then on eliminating x and y we get:
a_{1}(b_{2}c_{3} – b_{3}c_{2}) + b_{1}(c_{2}a_{3} – c_{3}a_{2}) + c_{1}(a_{2}b_{3} – a_{3}b_{2}) = 0
The expression on the left is called a determinant of the third order, and is denoted by:
A determinant of the third order consists of three rows and three columns.
The value of a determinant of order three is given by:
D = a_{1}(b_{2}c_{3} – b_{3}c_{2}) – b_{1}(a_{2}c_{3} – a_{3}c_{2}) + c_{1}(a_{2}b_{3} – a_{3}b_{2})
Using the Sarrus diagram to find the value of a thirdorder determinant:
Example 1: Find the value:
Solution:
= (27 + 42) – 2 (–36 –12) + 3 (28 – 6) = 231
Alternatively, Using Sarrus Diagram,
= (27 + 24 + 84) – (18 – 42 – 72)= 135 – (18 – 114) = 231
The minor of a given element of a determinant is the determinant obtained by deleting the row and the column in which the given element stands.
For example, the minor of a_{1} in
Hence a determinant of order three will have “9 minors”. If M_{ij} represents the minor of the element belonging to i^{th} row and j^{th} column then the cofactor of that element is given by : C_{ij} = (–1)^{i + j} . M_{ij.}
_{}
The sum of the products of the elements of any row (or column) of a matrix with their corresponding cofactors always equals the value of the determinant of the matrix.
This determinant $DD$ can be expressed in any of the following six forms:
Here, A_{i}, B_{i}, and $C\_i$C_{i} (for $i=1,2,3i\; =\; 1,\; 2,\; 3$) denote the cofactors of a_{i}, $b\_i$b_{i}, and $c\_i$c_{i} respectively.
Additionally, the sum of the products of the elements of any row (or column) with the cofactors of another row (or column) is always equal to zero. Consequently, we have:
$a2A1+b2B1+c2C1=0a\_2\; A\_1\; +\; b\_2\; B\_1\; +\; c\_2\; C\_1\; =\; 0$, $b1A1+b2A2+b3A3=0b\_1\; A\_1\; +\; b\_2\; A\_2\; +\; b\_3\; A\_3\; =\; 0$ and so forth.
In these equations, $A\_i$A_{i}, B_{i}, and C_{i} (for $i=1,2,3i\; =\; 1,\; 2,\; 3$) are again the cofactors of a_{i}, $b\_i$b_{i}, and $c\_i$c_{i} respectively.
(a) The value of a determinant remains unaltered, if the rows & 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.
(c) If all the elements of a row (or column) are zero, then the value of the determinant is zero.
(d) If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.
(e) If all the elements of a row (or column) are proportional (or identical) to the element of any other row, then the determinant vanishes, i.e. its value is zero.
Example: Prove that:
(f) If each element of any row (or column) is expressed as a sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants.
(g) Row  column operation : The value of a determinant remains unaltered under a column (C_{i} ) operation of the form C_{i} + aC_{j} + bC_{k} (j, k is not equal to i ) or row (R_{i} ) operation of the form R_{i} → R_{i} + aR_{j} + bR_{k} (j, k ≠ i). In other words, the value of a determinant is not altered by adding the elements of any row (or column) to the same multiples of the corresponding elements of any other row (or column).
(h) Factor theorem : If the elements of a determinant D are rational integral functions of x and two rows (or columns) become identical when x = a then (x – a) is a factor of D. Note that if r rows become identical when a is substituted for x, then (x – a)^{r–1} is a factor of D.
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 D1 is the determinant formed by replacing the elements of determinant D of order n by their corresponding cofactors then D_{1} = D^{n–1}
Example: Let a & b be the roots of equation ax^{2} + bx + c = 0 and S_{n} = a^{n} + b^{n} for n ≥ 1. Evaluate the value of the determinant
(i) Consistent Equations : Definite & unique solution (Intersecting lines)
(ii) Inconsistent Equations : No solution (Parallel lines)
(iii) Dependent Equations : Infinite solutions (Identical lines)
Note :
(i) If D ≠ 0 and atleast one of D_{1} , D_{2} , D_{3} ≠ 0, then the given system of equations is consistent and has unique non trivial solution.
(ii) If D ≠ 0 & D_{1} = D_{2} = D_{3} = 0, then the given system of equations is consistent and has trivial solution only.
(iii) If D = 0 but atleast one of D_{1} , D_{2} , D_{3} is not zero then the equations are inconsistent and have no solution.
(iv) If D = D_{1} = D_{2} = D_{3} = 0, then the given system of equations may have infinite or no solution.
Applications of Determinants:
209 videos443 docs143 tests

1. How do you find the value of a determinant? 
2. What are minors and cofactors of a determinant? 
3. How can a determinant be expanded in terms of the elements of any row or column? 
4. What are some properties of determinants? 
5. How does Cramer's Rule relate to determinants in solving a system of linear equations? 
209 videos443 docs143 tests


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