DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.


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1 Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of A If A b, then determinnt of A, denoted by A (or det A), is given by c d A = b c d = d bc Remrks (i) Only squre mtrices hve determinnts (ii) For mtrix A, A is red s determinnt of A nd not, s modulus of A 4 Determinnt of mtrix of order one Let A = [] be the mtrix of order, then determinnt of A is defined to be equl to 4 Determinnt of mtrix of order two b Let A = [ ij ] = c d be mtrix of order Then the determinnt of A is defined s: det (A) = A = d bc 43 Determinnt of mtrix of order three The determinnt of mtrix of order three cn be determined by expressing it in terms of second order determinnts which is known s expnsion of determinnt long row (or column) There re six wys of expnding determinnt of order 3 corresponding to ech of three rows (R, R nd R 3 ) nd three columns (C, C nd C 3 ) nd ech wy gives the sme vlue
2 66 MATHEMATICS Consider the determinnt of squre mtrix A = [ ij ] 3 3, ie, A Expnding A long C, we get A = ( ) ( ) ( ) = ( ) ( ) + 3 ( 3 3 ) Remrk In generl, if A = kb, where A nd B re squre mtrices of order n, then A = k n B, n =,, 3 44 Properties of Determinnts For ny squre mtrix A, A stisfies the following properties (i) A = A, where A = trnspose of mtrix A (ii) (iii) (iv) (v) (vi) If we interchnge ny two rows (or columns), then sign of the determinnt chnges If ny two rows or ny two columns in determinnt re identicl (or proportionl), then the vlue of the determinnt is zero Multiplying determinnt by k mens multiplying the elements of only one row (or one column) by k If we multiply ech element of row (or column) of determinnt by constnt k, then vlue of the determinnt is multiplied by k If elements of row (or column) in determinnt cn be expressed s the sum of two or more elements, then the given determinnt cn be expressed s the sum of two or more determinnts
3 DETERMINANTS 67 (vii) Notes: (i) (ii) (iii) If to ech element of row (or column) of determinnt the equimultiples of corresponding elements of other rows (columns) re dded, then vlue of determinnt remins sme If ll the elements of row (or column) re zeros, then the vlue of the determinnt is zero If vlue of determinnt Δ becomes zero by substituting x = α, then x α is fctor of Δ If ll the elements of determinnt bove or below the min digonl consists of zeros, then the vlue of the determinnt is equl to the product of digonl elements 45 Are of tringle Are of tringle with vertices (x, y ), (x, y ) nd (x 3, y 3 ) is given by x x x y y y Minors nd cofctors (i) Minor of n element ij of the determinnt of mtrix A is the determinnt obtined by deleting i th row nd j th column, nd it is denoted by M ij (ii) Cofctor of n element ij is given by A ij = ( ) i+j M ij (iii) Vlue of determinnt of mtrix A is obtined by the sum of products of elements of row (or column) with corresponding cofctors For exmple A = A + A + 3 A 3 (iv) If elements of row (or column) re multiplied with cofctors of elements of ny other row (or column), then their sum is zero For exmple, A + A + 3 A 3 = 0 47 Adjoint nd inverse of mtrix (i) The djoint of squre mtrix A = [ ij ] n n is defined s the trnspose of the mtrix
4 68 MATHEMATICS [ ij ] n n, where A ij is the cofctor of the element ij It is denoted by dj A 3 A A A3 If A 3, then dj A A A A 3, where A ij is cofctor of ij A A A (ii) A (dj A) = (dj A) A = A I, where A is squre mtrix of order n (iii) A squre mtrix A is sid to be singulr or nonsingulr ccording s A = 0 or A 0, respectively (iv) If A is squre mtrix of order n, then dj A = A n (v) If A nd B re nonsingulr mtrices of the sme order, then AB nd BA re lso nonsingulr mtrices of the sme order (vi) The determinnt of the product of mtrices is equl to product of their respective determinnts, tht is, AB = A B (vii) If AB = BA = I, where A nd B re squre mtrices, then B is clled inverse of A nd is written s B = A Also B = (A ) = A (viii) A squre mtrix A is invertible if nd only if A is nonsingulr mtrix (ix) If A is n invertible mtrix, then A = 48 System of liner equtions (dj A) A (i) Consider the equtions: x + b y + c z = d x + b y + c z = d 3 x + b 3 y + c 3 z = d 3, In mtrix form, these equtions cn be written s A X = B, where A = b c x d b c,x y ndb d b c z d (ii) Unique solution of eqution AX = B is given by X = A B, where A 0
5 DETERMINANTS 69 (iii) (iv) A system of equtions is consistent or inconsistent ccording s its solution exists or not For squre mtrix A in mtrix eqution AX = B () If A 0, then there exists unique solution (b) If A = 0 nd (dj A) B 0, then there exists no solution (c) If A = 0 nd (dj A) B = 0, then system my or my not be consistent 4 Solved Exmples Short Answer (SA) Exmple If x x 8 3, then find x Solution We hve x This gives 8 x 8 3 x 40 = 8 40 x = 9 x = ± 3 Exmple If x x Δ= y y, Δ = yz zx xy, then prove tht Δ + Δ = 0 z z x y z Solution We hve yz zx xy x y z Interchnging rows nd columns, we get yz zx xy x y z x xyz x = y xyz y xyz z xyz z
6 70 MATHEMATICS = x x xyz y y xyz Interchnging C z nd C z = x ( ) y y z x z Δ + Δ = 0 Exmple 3 Without expnding, show tht cosec cot cot cosec 4 40 = 0 Solution Applying C C C C 3, we hve cosec cot cot cot cosec cosec 0 40 = 0 cot θ 0 cosec θ = Exmple 4 Show tht x p q p x q q q x = (x p) (x + px q ) Solution Applying C C C, we hve x p p q p x x q 0 q x p q ( x p) x q 0 q x
7 DETERMINANTS 7 Expnding long C, we hve 0 p + x q = ( x p) x q Applying R R + R 0 q x ( x p)( px x q ) = ( x p)( x px q ) Exmple 5 If 0 b c b 0 c b c b c 0, then show tht is equl to zero Solution Interchnging rows nd columns, we get 0 b c b 0 b c c c b 0 Tking common from R, R nd R 3, we get 0 b c 3 ( ) b 0 c b c b c 0 = 0 or = 0 Exmple 6 Prove tht (A ) = (A ), where A is n invertible mtrix Solution Since A is n invertible mtrix, so it is nonsingulr We know tht A = A But A 0 So A 0 ie A is invertible mtrix Now we know tht AA = A A = I Tking trnspose on both sides, we get (A ) A = A (A ) = (I) = I Hence (A ) is inverse of A, ie, (A ) = (A ) Long Answer (LA) Exmple 7 If x = 4 is root of x 3 x = 0, then find the other two roots 3 x
8 7 MATHEMATICS Solution Applying R (R + R + R 3 ), we get x 4 x 4 x 4 x 3 x Tking (x + 4) common from R, we get ( x 4) x 3 x Applying C C C, C 3 C 3 C, we get 0 0 ( x 4) x 0 3 x 3 Expnding long R, Δ = (x + 4) [(x ) (x 3) 0] Thus, Δ = 0 implies x = 4,, 3 Exmple 8 In tringle ABC, if sina sinb sinc 0 sina +sin A sinb+sin B sinc+sin C, then prove tht ΔABC is n isoceles tringle Solution Let Δ = sina sinb sinc sina +sin A sinb+sin B sinc+sin C
9 DETERMINANTS 73 = sina sinb sinc cos A cos B cos C R 3 R 3 R = 0 0 sin A sin B sin A sin C sin B (C 3 C 3 C nd C C C ) cos A cos A cos B cos B cos C Expnding long R, we get Δ = (sinb sina) (sin C sin B) (sinc sin B) (sin B sin A) = (sinb sina) (sinc sinb) (sinc sin A) = 0 either sinb sina = 0 or sinc sinb or sinc sina = 0 A = B or B = C or C = A ie tringle ABC is isoceles Exmple 9 Show tht if the determinnt 3 sin3 7 8 cos 0, then sinθ = 0 or 4 Solution Applying R R + 4R nd R 3 R 3 + 7R, we get 3 sin3 5 0 cos 4sin sin3 or [5 ( + 7 sin3θ) 0 (cosθ + 4sin3θ)] = 0 or + 7sin3θ cosθ 8sin3θ = 0 or cos θ sin 3θ = 0 sinθ (4sin θ + 4sinθ 3) = 0
10 74 MATHEMATICS or sinθ = 0 or (sinθ ) = 0 or (sinθ + 3) = 0 or sinθ = 0 or sinθ = (Why?) Objective Type Questions Choose the correct nswer from the given four options in ech of the Exmple 0 nd Exmple 0 Let Ax x A B C By y nd x y z Cz z zy zx xy, then (A) Δ = Δ (B) Δ Δ (C) Δ Δ = 0 (D) None of these Solution (C) is the correct nswer since A B C x y z zy zx xy A = B C x y z yz zx xy = Ax x xyz B y y xyz xyz Cz z xyz = xyz xyz Ax x By y Cz z = Δ Exmple If x, y R, then the determinnt in the intervl cos x sin x sin x cos x cos( x y) sin( x y) 0 lies (A), (B) [, ] (C), (D),, Solution The correct choice is A Indeed pplying R 3 R 3 cosyr + sinyr, we get
11 DETERMINANTS 75 cos x sin x sin x cos x 0 0 siny cosy Expnding long R 3, we hve Δ = (siny cosy) (cos x + sin x) = (siny cosy) = siny cosy π = cos siny sin cosy = 4 4 sin (y ) 4 Hence Δ Fill in the blnks in ech of the Exmples to 4 Exmple If A, B, C re the ngles of tringle, then sin A cota sin B cotb sin C cotc Solution Answer is 0 Apply R R R, R 3 R 3 R Exmple 3 The determinnt Δ= is equl to Solution Answer is 0Tking 5 common from C nd C 3 nd pplying C C 3 3 C, we get the desired result Exmple 4 The vlue of the determinnt
12 76 MATHEMATICS sin 3 sin 67 cos80 sin 67 sin 3 cos 80 cos80 sin 3 sin 67 Solution Δ = 0 Apply C C + C + C 3 Stte whether the sttements in the Exmples 5 to 8 is True or Flse Exmple 5 The determinnt cos( x y) sin ( x y) cos y sin x cos x sin y cos x sin x cos y is independent of x only Solution True Apply R R + sinyr + cosy R 3, nd expnd Exmple 6 The vlue of n n+ n+ 4 C C C n n+ n+ 4 C C C is 8 Solution True Exmple 7 If x 5 A y 3 z, xyz = 80, 3x + y + 0z = 0, then A dj A Solution : Flse
13 DETERMINANTS 77 Exmple 8 If A x, A 3 3 y then x =, y = Solution True 43 EXERCISE Short Answer (SA) Using the properties of determinnts in Exercises to 6, evlute: x x x x x x y z x y z x y z 3 0 xy x y 0 xz zy xz yz 0 4 3x x y x z x y 3y z y x z y z 3z 5 x 4 x x x x 4 x x x x 4 6 b c b b c b c c c b Using the proprties of determinnts in Exercises 7 to 9, prove tht: 7 yz yz y z z x zx z x x y xy x y 0 8 y z z y z z x x 4xyz y x x y
14 78 MATHEMATICS 9 ( ) If A + B + C = 0, then prove tht cosc cosb cosc cosa 0 cosb cos A If the coordintes of the vertices of n equilterl tringle with sides of length re (x, y ), (x, y ), (x 3, y 3 ), then y 4 x x x y y = 4 Find the vlue of θ stisfying sin3θ 4 3 cosθ = If 4 x 4 x 4 x 4 x 4 x 4 x 0, then find vlues of x 4 x 4 x 4 x 4 If,, 3,, r re in GP, then prove tht the determinnt r r 5 r 9 r 7 r r 5 r r 7 r is independent of r 5 Show tht the points ( + 5, 4), (, + 3) nd (, ) do not lie on stright line for ny vlue of 6 Show tht the ΔABC is n isosceles tringle if the determinnt
15 DETERMINANTS 79 Δ= + cosa + cosb + cosc = 0 cos A + cos A cos B + cos B cos C+ cos C 7 Find A if 0 A 0 0 nd show tht A 3I A Long Answer (LA) 8 If 0 A=, find A 0 Using A, solve the system of liner equtions x y = 0, x y z = 8, y + z = 7 9 Using mtrix method, solve the system of equtions 3x + y z = 3, x + y + 3z = 6, x y + z = Given A 4 4, B 3 4, find BA nd use this to solve the 5 0 system of equtions y + z = 7, x y = 3, x + 3y + 4z = 7 b c If + b + c 0 nd b c = 0, then prove tht = b = c c b Prove tht quotient bc c b b c c b b c bc b c bc c b is divisible by + b + c nd find the
16 80 MATHEMATICS 3 If x + y + z = 0, prove tht x yb zc b c yc z xb = xyz c b zb xc y b c Objective Type Questions (MCQ) Choose the correct nswer from given four options in ech of the Exercises from 4 to 37 4 If x x 7 3, then vlue of x is (A) 3 (B) ± 3 (C) ± 6 (D) 6 5 The vlue of determinnt b b+ c b c+ b c + b c (A) 3 + b 3 + c 3 (B) 3 bc (C) 3 + b 3 + c 3 3bc (D) none of these 6 The re of tringle with vertices ( 3, 0), (3, 0) nd (0, k) is 9 sq units The vlue of k will be (A) 9 (B) 3 (C) 9 (D) 6 7 The determinnt b b b c bc c b b b b bc c c b equls (A) bc (b c) (c ) ( b) (B) (b c) (c ) ( b) (C) ( + b + c) (b c) (c ) ( b) (D) None of these
17 DETERMINANTS 8 8 The number of distinct rel roots of sin x cos x cos x cos x sin x cos x 0 cos x cos x sin x in the intervl π π x is 4 4 (A) 0 (B) (C) (D) 3 9 If A, B nd C re ngles of tringle, then the determinnt cosc cosb cosc cosa is equl to cosb cos A (A) 0 (B) (C) (D) None of these 30 Let f (t) = cost t sint t t sin t t t f () t, then lim t 0 t is equl to (A) 0 (B) (C) (D) 3 3 The mximum vlue of sin cos is (θ is rel number) (A) (B) 3 (C) (D) 3 4
18 8 MATHEMATICS 3 If f (x) = 0 x x b x 0 x c x b x c 0, then (A) f () = 0 (B) f (b) = 0 (C) f (0) = 0 (D) f () = 0 33 If A = , then A exists if (A) λ = (B) λ (C) λ (D) None of these 34 If A nd B re invertible mtrices, then which of the following is not correct? (A) dj A = A A (B) det(a) = [det (A)] (C) (AB) = B A (D) (A + B) = B + A 35 If x, y, z re ll different from zero nd x + y + z is x y 0, then vlue of z (A) x y z (B) x y z (C) x y z (D) 36 The vlue of the determinnt x x y x y x y x x y x y x y x is (A) 9x (x + y) (B) 9y (x + y) (C) 3y (x + y) (D) 7x (x + y)
19 DETERMINANTS There re two vlues of which mkes determinnt, Δ = = 86, then 0 4 sum of these number is (A) 4 (B) 5 (C) 4 (D) 9 Fill in the blnks 38 If A is mtrix of order 3 3, then 3A = 39 If A is invertible mtrix of order 3 3, then A x x x x 40 If x, y, z R, then the vlue of determinnt equl to x x x x x x x x is 4 If cosθ = 0, then 0 cosθ sinθ cosθ sin θ 0 = sin θ 0 cosθ 4 If A is mtrix of order 3 3, then (A ) = 43 If A is mtrix of order 3 3, then number of minors in determinnt of A re 44 The sum of the products of elements of ny row with the cofctors of corresponding elements is equl to 45 If x = 9 is root of x 3 7 x 7 6 x = 0, then other two roots re 46 0 xyz x z y x 0 y z z x z y 0 =
20 84 MATHEMATICS 47 If f (x) = ( + x) ( + x) ( + x) ( + x) ( + x) ( + x) ( + x) ( + x) ( + x) = A + Bx + Cx +, then A = Stte True or Flse for the sttements of the following Exercises: 48 3 A = 3 A, where A is squre mtrix nd A 0 49 (A) = A, where is ny rel number nd A is squre mtrix 50 A A, where A is nonsingulr mtrix 5 If A nd B re mtrices of order 3 nd A = 5, B = 3, then 3AB = = If the vlue of third order determinnt is, then the vlue of the determinnt formed by replcing ech element by its cofctor will be x+ x+ x+ x+ x+ 3 x+ b = 0, where, b, c re in AP x+ 3 x+ 4 x+ c 54 dj A = A, where A is squre mtrix of order two 55 The determinnt sin A cos A sin A +cosb sinb cosa sinb+cosb sin C cos A sin C+cosB is equl to zero x p u l f 56 If the determinnt y b q v m g splits into exctly K determinnts of z c r+ w n h order 3, ech element of which contins only one term, then the vlue of K is 8
21 DETERMINANTS 85 p x p+ x + x + p 57 Let b q y 6, then Δ = q+ y b+ y b+ q = 3 c r z r+ z c+ z c+ r 58 The mximum vlue of ( sin ) is cos
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