Chapter-8-DETERMINANT-(E)-01-THEORY

8.1.1 Definition . (1) Consider two equations, .....(i) and .....(ii) Multiplying (i) by and (ii) by and subtracting, dividing by x, we get, The result is represented by Which is known as determinant of order two and is the expansion of this determinant. The horizontal lines are called rows and vertical lines are called columns. Now let us consider three homogeneous linear equations , and Eliminated x, y, z from above three equations we obtain .....(iii) The L.H.S. of (iii) is represented by Its contains three rows and three columns, it is called a determinant of third order. Note :  The number of elements in a second order is and the number of elements in a third order determinant is . (2) Rows and columns of a determinant : In a determinant horizontal lines counting from top 1st, 2nd, 3rd,….. respectively known as rows and denoted by and vertical lines counting left to right, 1st, 2nd, 3rd,….. respectively known as columns and denoted by (3) Shape and constituents of a determinant : Shape of every determinant is square. If a determinant of n order then it contains n rows and n columns. i.e., Number of constituents in determinants = n2 (4) Sign system for expansion of determinant : Sign system for order 2, order 3, order 4,….. are given by , 8.1.2 Expansion of Determinants . Unlike a matrix, determinant is not just a table of numerical data but (quite differently) a short hand way of writing algebraic expression, whose value can be computed when the values of terms or elements are known. (1) The 4 numbers arranged as is a determinant of second order. These numbers are called elements of the determinant. The value of the determinant is defined as . The expanded form of determinant has 2! terms. (2) The 9 numbers arranged as is a determinant of third order. Take any row (or column); the value of the determinant is the sum of products of the elements of the row (or column) and the corresponding determinant obtained by omitting the row and the column of the element with a proper sign, given by the rule , where i and j are the number of rows and the number of columns respectively of the element of the row (or the column) chosen. Thus = The diagonal through the left-hand top corner which contains the element is called the leading diagonal or principal diagonal and the terms are called the leading terms. The expanded form of determinant has 3! terms. Short cut method or Sarrus diagram method : To find the value of third order determinant, following method is also useful Taking product of R.H.S. diagonal elements positive and L.H.S. diagonal elements negative and adding them. We get the value of determinant as Note : This method does not work for determinants of order greater than three. 8.1.3 Evaluation of Determinants . If A is a square matrix of order 2, then its determinant can be easily found. But to evaluate determinants of square matrices of higher orders, we should always try to introduce zeros at maximum number of places in a particular row (column) by using the properties and then we should expand the determinant along that row (column). We shall be using the following notations to evaluate a determinant : (1) to denote row. (2) to denote the interchange of and rows. (3) to denote the addition of times the elements of row to the corresponding elements of row. (4) to denote the multiplication of all element of row by . Similar notations are used to denote column operations if R is replaced by C. 8.1.4 Properties of Determinants . P-1 : The value of determinant remains unchanged, if the rows and the columns are interchanged. If and . Then D and are transpose of each other. Note :  Since the determinant remains unchanged when rows and columns are interchanged, it is obvious that any theorem which is true for ‘rows’ must also be true for ‘columns’. P-2 : If any two rows (or columns) of a determinant be interchanged, the determinant is unaltered in numerical value but is changed in sign only. Let and . Then P-3 : If a determinant has two rows (or columns) identical, then its value is zero. Let . Then, D = 0 P-4 : If all the elements of any row (or column) be multiplied by the same number, then the value of determinant is multiplied by that number. Let and . Then P-5 : If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of the determinants. e.g., = + P-6 : 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., and . Then Note :  It should be noted that while applying P-6 at least one row (or column) must remain unchanged. P-7 : If all elements below leading diagonal or above leading diagonal or except leading diagonal elements are zero then the value of the determinant equal to multiplied of all leading diagonal elements. e.g., P-8 : If a determinant D becomes zero on putting , then we say that is factor of determinant. e.g., if . At (because and are identical at ) Hence is a factor of D. Note :  It should be noted that while applying operations on determinants then at least one row (or column) must remain unchanged.or, Maximum number of operations = order or determinant –1  It should be noted that if the row (or column) which is changed by multiplied a non zero number, then the determinant will be divided by that number. Example: 1 If and 1, , are the cube roots of unity, then has the value (a) 0 (b) (c) (d) 1 Solution: (a) Applying , we get Example: 2 (a) (b) (c) (d) Solution: (a) Trick : Put and , then . option (a) gives Example: 3 The value of is equal to zero, where m is []0 (a) 6 (b) 4 (c) 5 (d) None of these Solution: (c) = 0 Applying  = 0 Clearly satisfies the above result Example: 4 If are in G.P. then the value of the determinant is (a) –2 (b) 1 (c) 2 (d) 0 Solution: (d) are in G.P. Putting these values in the second column of the given determinant, we get Example: 5 The value of (a) 0 (b) (c) (d) None of these Solution: (a) Applying Trick : Putting , we get option (a) is correct Example: 6 If x, y, z are integers in A.P. lying between 1 and 9 and x51, y41 and z31 are three digit numbers then the value of is (a) (b) (c) 0 (d) None of these Solution: (c) , Applying x, y, z are in A.P. , , Example: 7 If , the value of x which satisfies the equation is (a) (b) (c) (d) Solution: (a) Expanding determinant, we get, Either or . Since satisfies the given equation. Trick : On putting , we observe that the determinant becomes is a root of the given equation. Example: 8 The number of distinct real roots of in the interval is (a) 0 (b) 2 (c) 1 (d) 3 Solution: (c) Applying, and  But in . Hence Example: 9 If , then value of t is (a) 16 (b) 18 (c) 17 (d) 19 Solution: (b) Since it is an identity in so satisfied by every value of . Now put in the given equation, we have Example: 10 If , then is equal to (a) 0 (b) 1 (c) 100 (d) –100 Solution: (a) Applying , we get . Hence Example: 11 The value of is (a) 6! (b) 5! (c) (d) None of these Solution: (b) The elements in the leading diagonal are 1, 2, 3, 4, 5. On one side of the leading diagonal all the elements are zero. The value of the determinant = The product of the elements in the leading diagonal = 1.2.3.4.5 = 5! Example: 12 The determinant if (a) a, b, c are in A.P. (b) a, b, c are in G.P. or is a factor of (c) a, b, c are in H.P. (d) is a root of the equation Solution: (b) Applying , we get or is a root of or a, b, c are in G.P. is a factor of or a, b, c are in G.P. 8.1.5 Minors and Cofactors . (1) Minor of an element : If we take the element of the determinant and delete (remove) the row and column containing that element, the determinant left is called the minor of that element. It is denoted by Consider the determinant , then determinant of minors , where minor of , minor of minor of Similarly, we can find the minors of other elements . Using this concept the value of determinant can be or, or, . (2) Cofactor of an element : The cofactor of an element (i.e. the element in the row and column) is defined as times the minor of that element. It is denoted by or or . If , then determinant of cofactors is , where , and Similarly, we can find the cofactors of other elements. Note : The sum of products of the element of any row with their corresponding cofactor is equal to the value of determinant i.e. where the capital letters etc. denote the cofactors of etc.  In general, it should be noted or  If is the determinant formed by replacing the elements of a determinant by their corresponding cofactors, then if , then , , where n is the order of the determinant. Example: 13 The cofactor of the element 4 in the determinant is (a) 4 (b) 10 (c) –10 (d) –4 Solution: (b) The cofactor of element 4, in the 2nd row and 3rd column is = Example: 14 If and denote the cofactors of respectively, then the value of the determinant is (a) (b) (c) (d) Solution: (b) We know that . Trick : According to property of cofactors Example: 15 If the value of a third order determinant is 11, then the value of the square of the determinant formed by the cofactors will be (a) 11 (b) 121 (c) 1331 (d) 14641 Solution: (d) . But we have to find the value of the square of the determinant, so required value is . 8.1.6 Product of two Determinants . Let the two determinants of third order be, and . Let D be their product. (1) Method of multiplying (Row by row) : Take the first row of and the first row of i.e. and multiplying the corresponding elements and add. The result is is the first element of first row of D. Now similar product first row of and second row of gives is the second element of first row of D, and the product of first row and third row of gives is the third element of first row of D. The second row and third row of D is obtained by multiplying second row and third row of with 1st , 2nd , 3rd row of , in the above manner. Hence, Note :  We can also multiply rows by columns or columns by rows or columns by columns. Example: 16 For all values of A, B, C and P, Q, R the value of is (a) 0 (b) (c) (d) Solution: (a) The determinant can be expanded as Example: 17 (a) 7 (b) 10 (c) 13 (d) 17 Solution: (b) 8.1.7 Summation of Determinants . Let , where a, b, c, l, m and n are constants, independent of r. Then, . Here function of r can be the elements of only one row or one column. Example: 18 If , then (a) 0 (b) 25 (c) 625 (d) None of these Solution: (d) , , , , Example: 19 The value of , if is (a) 0 (b) 1 (c) –1 (d) None of these Solution: (a) Applying = 0 [ and are identical] 8.1.8 Differentiation and Integration of Determinants. (1) Differentiation of a determinant : (i) Let be a determinant of order two. If we write , where and denote the 1st and 2nd columns, then where denotes the column which contains the derivative of all the functions in the column . In a similar fashion, if we write , then (ii) Let be a determinant of order three. If we write , then and similarly if we consider , then (iii) If only one row (or column) consists functions of x and other rows (or columns) are constant, viz. Let , then and in general where n is any positive integer and denotes the derivative of . Example: 20 If and are the given determinants, then (a) (b) (c) (d) Solution: (b) and Example: 21 If , then the value of the determinant , where is (a) (b) (c) (d) None of these Solution: (d) Taking common from and becomes identical. Hence the value of determinant is zero. (2) Integration of a determinant : Let , where a, b, c, l, m and n are constants. Note : If the elements of more than one column or rows are functions of x then the integration can be done only after evaluation/expansion of the determinant. Example: 22 If , then is equal to (a) 1/4 (b) 1/2 (c) 0 (d) –1/2 Solution: (d) Applying 8.1.9 Application of Determinants in solving a system of Linear Equations. Consider a system of simultaneous linear equations is given by .....(i) A set of values of the variables x, y, z which simultaneously satisfy these three equations is called a solution. A system of linear equations may have a unique solution or many solutions, or no solution at all, if it has a solution (whether unique or not) the system is said to be consistent. If it has no solution, it is called an inconsistent system. If in (i) then the system of equations is said to be a homogeneous system. Otherwise it is called a non-homogeneous system of equations. Theorem 1 : (Cramer’s rule) The solution of the system of simultaneous linear equations .....(i) and .....(ii) is given by , where , and , provided that Note :  Here is the determinant of the coefficient matrix . The determinant is obtained by replacing first column in D by the column of the right hand side of the given equations. The determinant is obtained by replacing the second column in D by the right most column in the given system of equations. (1) Solution of system of linear equations in three variables by Cramer’s rule : Theorem 2 : (Cramer’s Rule) The solution of the system of linear equations .....(i) .....(ii) .....(iii) is given by and , where and Provided that Note : Here D is the determinant of the coefficient matrix. The determinant is obtained by replacing the elements in first column of D by is obtained by replacing the element in the second column of D by and to obtain , replace elements in the third column of D by . Theorem 3 : (Cramer’s Rule) Let there be a system of n simultaneous linear equation n unknown given by Let and let , be the determinant obtained from D after replacing the column by . Then, , Provided that (2) Conditions for consistency Case 1 : For a system of 2 simultaneous linear equations with 2 unknowns (i) If , then the given system of equations is consistent and has a unique solution given by . (ii) If and , then the system is consistent and has infinitely many solutions. (iii) If and one of and is non-zero, then the system is inconsistent. Case 2 : For a system of 3 simultaneous linear equations in three unknowns (i) If , then the given system of equations is consistent and has a unique solution given by and (ii) If and , then the given system of equations is consistent with infinitely many solutions. (iii) If and at least one of the determinants is non-zero, then given of equations is inconsistent. (3) Algorithm for solving a system of simultaneous linear equations by Cramer’s rule (Determinant method) Step 1 : Obtain and Step 2 : Find the value of D. If , then the system of the equations is consistent has a unique solution. To find the solution, obtain the values of and . The solutions is given by and . If go to step 3. Step 3 : Find the values of . If at least one of these determinants is non-zero, then the system is inconsistent. If then go to step 4 Step 4 : Take any two equations out of three given equations and shift one of the variables, say z on the right hand side to obtain two equations in x, y. Solve these two equations by Cramer’s rule to obtain x, y, in terms of z. Note:  The system of following homogeneous equations , , is always consistent. If , then this system has the unique solution known as trivial solution. But if , then this system has an infinite number of solutions. Hence for non-trivial solution . Example: 23 If the system of linear equations has a non-zero solution, then a, b, c (a) Are in A.P. (b) Are in G.P. (c) Are in H.P. (d) Satisfy Solution: (c) System of linear equations has a non-zero solution, then Applying ; Applying . On simplification ; a, b, c are in H.P. Example: 24 If the system of equations and has infinite solutions, then the value of a is (a) –1 (b) 1 (c) 0 (d) No real values Solution: (a) Example: 25 If the system of equations and , where has a non-trivial solution, then the value of is [] (a) –1 (b) 0 (c) 1 (d) None of these Solution: (c) As the system of the equations has a non-trivial solution Applying and Example: 26 If the system of equations is inconsistent, then the value of k is (a) –3 (b) (c) 0 (d) 2 Solution: (a) For the equations to be inconsistent and , Hence system is inconsistent for . Example: 27. The equations give infinite number of values of the triplet (x, y, z) if [] (a) (b) (c) (d) None of these Solution: (c) Each of the first three options contains . When , the last two equations become and . Obviously, when these equations become the same. So we are left with only two independent equations to find the values of the three unknowns. Consequently, there will be infinite solutions. Example: 28 The value of for which the system of equations , has no solution is (a) 3 (b) –3 (c) 2 (d) –2 Solution: (d) . For no solution the necessary condition is . It can be see that for , there is no solution for the given system of equations. 8.1.10 Application of Determinants in Co-ordinate Geometry. (1) Area of triangle whose vertices are is (2) If are the sides of a triangle, then the area of the triangle is given by , where are the cofactors of the elements respectively in the determinant . (3) The equation of a straight line passing through two points and is (4) If three lines are concurrent if (5) If represents a pair of straight lines then (6) The equation of circle through three non-collinear points is Example: 29 The three lines are concurrent only when (a) (b) (c) (d) None of these Solution: (a, b) Three lines are concurrent if or, Also, or . 8.1.11 Some Special Determinants. (1) Symmetric determinant : A determinant is called symmetric determinant if for its every element e.g., (2) Skew-symmetric determinant : A determinant is called skew symmetric determinant if for its every element e.g., Note :  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. For example (i) (ii) (Perfect square) (iii) (3) Cyclic order : If elements of the rows (or columns) are in cyclic order. i.e. (i) (ii) (iii) (iv) (v) Note :  These results direct applicable in lengthy questions (As behavior of standard results) Example: 30 is equal to (a) 100 (b) 500 (c) 1000 (d) 0 Solution: (d) This determinant of skew symmetric of odd order, hence is equal to 0. Example: 31 (a) (b) (c) (d) None of these Solution: (c) Applying ; Applying Example: 32 If Where a, b, c are all different, then the determinant vanishes when [] (a) (b) (c) (d) Solution: (b) …… (i) Now,  Applying or Example: 33 If and are the roots of the equations then value of the determinant is (a) p (b) q (c) (d) 0 Solution: (d) Since are the roots of , Applying , We get, = Example: 34 If ,then (a) (–3, –9) (b) (–5, –9) (c) (–3, –5) (d) (3, –9) Solution: (b) Putting in both sides, we get, by expansion. is the coefficient of x or constant term in the differentiation of determinant. Differentiate both sides, Putting both sides, we get ; .