Determinants 367 kho tài liệu bách khoa

KEY CONCEPTS DETERMINANT The symbol is called the determinant of order two Its value is given by : D = a1 b2 − a2 b1 The symbol is called the determinant of order three − a2 Its value can be found as : D = a1 D = a1 − b1 + c1 + a3 OR and so on In this manner we can expand a determinant in ways using elements of ; R1 , R2 , R3 or C1 , C2 , C3 Following examples of short hand writing large expressions are : (i) The lines : a1x + b1y + c1 = (1) a2x + b2y + c2 = (2) a3x + b3y + c3 = (3) =0 are concurrent if , (ii) Condition for the consistency of three simultaneous linear equations in variables ax² + hxy + by² + gx + fy + c = represents a pair of straight lines if : abc + fgh − af² − bg² − ch² = = (iii) Area of a triangle whose vertices are (xr , yr) ; r = , , is : D= (iv) Equation of a straight line passsing through (x1 , y1) & (x2 , y2) is =0 MINORS : The minor of a given element of a 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 a1 in (Key Concept 2) is If D = then the three points are collinear & the minor of b2 is Hence a determinant of order two will have “4 minors” & a determinant of order three will have “9 minors” COFACTOR : If Mij represents the minor of some typical element then the cofactor is defined as : Cij = (−1)i+j Mij ; Where i & j denotes the row & column in which the particular element lies Note that the value of a determinant of order three in terms of ‘Minor’ & ‘Cofactor’ can be written as : D = a11M11 − a12M12 + a13M13 OR D = a11C11 + a12C12 + a13C13 & so on ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) PROPERTIES OF DETERMINANTS : − : The value of a determinant remains unaltered , if the rows & columns are inter changed P− = e.g if D = = D′ D & D′ are transpose of each other If D′ = − D then it is SKEW SYMMETRIC determinant but D′ = D ⇒ D = ⇒ D = ⇒ Skew symmetric determinant of third order has the value zero − 2: If any two rows (or columns) of a determinant be interchanged , the value of determinant is P− changed in sign only e.g Let D = Then D′ = − D & D′ = − 3: If a determinant has any two rows (or columns) identical , then its value is zero P− then it can be verified that D = e.g Let D = P− − 4: If all the elements of any row (or column) be multiplied by the same number, then the determinant is multiplied by that number e.g P− −5: and If D = D′ = Then D′= KD 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 two 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 Let D = D′ = and + + + + + + Then D′ = D Note : that while applying this property ATLEAST ONE ROW (OR COLUMN) must remain unchanged P− 7: If by putting x = a the value of a determinant vanishes then (x − a) is a factor of the determinant MULTIPLICATION OF TWO DETERMINANTS : (i) = + + + + Similarly two determinants of order three are multiplied ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) (ii) ≠ then , D² = If D = Consider PROOF : where Ai , Bi , Ci are cofactors × = Note : a1A2 + b1B2 + c1C2 = etc = D3 ⇒ therefore , D x = D² OR SYSTEM OF LINEAR EQUATION (IN TWO VARIABLES) : (i) Consistent Equations : Definite & unique solution [ intersecting lines ] (ii) Inconsistent Equation : No solution [ Parallel line ] (iii) Dependent equation : Infinite solutions [ Identical lines ] Let a1x + b1y + c1 = & = D² & a2x + b2y + c2 = then : = ≠ ⇒ Given equations are inconsistent = = ⇒ Given equations are dependent CRAMER'S RULE : [ Simultaneous Equations Involving Three Unknowns ] Let ,a1x + b1y + c1z = d1 (I) ; a2x + b2y + c2z = d2 (II) ; a3x + b3y + c3z = d3 (III) Then , x= Where D= , Y= , Z= ; D1 = ; D2 = & D3 = NOTE : (a) If D ≠ and alteast one of D1 , D2 , D3 ≠ , then the given system of equations are consistent and have unique non trivial solution (b) If D ≠ & D1 = D2 = D3 = , then the given system of equations are consistent and have trivial solution only (c) If D = D1 = D2 = D3 = , then the given system of equations are consistent and have infinite solutions In case + + + + + + = = =   represents these parallel planes then also  (d) D = D1 = D2 = D3 = but the system is inconsistent If D = but atleast one of D1 , D2 , D3 is not zero then the equations are inconsistent and have no solution 10 If x , y , z are not all zero , the condition for a1x + b1y + c1z = ; a2x + b2y + c2z = & a3x + b3y + c3z = to be consistent in x , y , z is that = Remember that if a given system of linear equations have Only Zero Solution for all its variables then the given equations are said to have TRIVIAL SOLUTION ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) EXERCISE–I − Q.1 (a) + − Prove that the value of the determinant + + (b) Q.2 − is real − On which one of the parameter out of a, p, d or x, the value of the determinant − ! + ! − ! + ! does not depend Without expanding as far as possible, prove that + + (a) + + + + + = (a − 1)3 = [(x−y) (y−z) (z−x) (x+y+z)] (b) = and x , y , z are all different then , prove that xyz = − Q.3 If Q.4 Using properties of determinants or otherwise evaluate Q.5 Prove that Q.6 If D = Q.7 Prove that Q.8 Prove that Q.9 # Show that the value of the determinant # # − − − − + + + + + + + − + + + then prove that D′ = D − − + − + − = (a + b + c)3 − − and D′ = − + + − − − = (1 + a² + b²)3 = (a + b + c) (a² + b² + c²) +%! # +$! # +" ! # +%! # +$! # +" ! # +%! +$! vanishes for all values of +" ! A, B, C, P, Q & R where A + B + C + P + Q + R = ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.10 Q.11 (β + γ − α − δ) (γ + α − β − δ ) (α + β − γ − δ) Prove that (β + γ − α − δ) (γ + α − β − δ ) (α + β − γ − δ) = − 64(α − β)(α − γ)(α − δ)(β − γ) (β − δ) (γ− δ) + !& + !& + !& & + !& + !& For a fixed positive integer n, if D= + !&   + !& then show that  −  is divisible   &! + !& by n Q.12 Solve for x + + + (a) + + + + + + = − − − (b) − − If a + b + c = , solve for x : Q.14 If a2 + b2 + c2 = then show that the value of the determinant Q.15 Q.16 + ! θ − θ! − θ! − θ! + ! − θ! + If p + q + r = , prove that ( ) − θ ( ) + − − ' = −' = Q.13 + − − − − θ! − θ! + ! simplifies to cos2θ θ ) = pqr ( − − − If a , b , c are all different & = 0, then prove that, abc(ab + bc + ca) = a + b + c +λ Q.17 Show that, Q.18 Prove that : Q.19 Q.20 +λ +λ + ! − ! + ! − ! is divisible by λ2 and find the other factor + ! − ! = # +# In a D ABC, determine condition under which # Prove that : − ! − ! − (! − (! − )! − )! − ! − (! − )! = # # +# # + + ! ! + (! + (! + )! + )! + ! + (! + )! # +# ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) =0 Q.21 Q.22 Q.23 − ! − ! − ! − ! − ! − ! − ! − ! − ! Prove that = 2(a1− a2)(a2− a3)(a3− a1)(b1− b2)(b2− b3)(b3− b1) α+β+γ +δ αβ + γδ Prove that α + β + γ + δ α + β ! γ + δ! αβ γ + δ! + γδ α + β! = αβ + γδ αβ γ + δ! + γδ α + β! αβγδ ax1² + by1² + cz12 = ax22 + by22 + cz22 = ax32 + by32 + cz32 = d ax2x3 + by2y3 + cz2z3 = ax3x1 + by3y1 + cz3z1 = ax1x2 + by1y2 + cz1z2 = f, If and  + = (d − f)   then prove that αr + βr + + then show that + + γr + + + *    (a , b , c ≠ 0) + + + = (α − β)2 (β − γ)2 (γ − α)2 Q.24 If Sr = Q.25 If u = ax2 + bxy + cy2 , u′ = a′x2 + b′xy + c′y2 Prove that − ′ ′ ′ + ′ + ′ = + =− ′ + ′ , + ,′ ′ + ′ EXERCISE–II Q.1 Solve the following using Cramer’s rule and state whether consistent or not (a) + + −' = + − − = + − + = (b) + + = + + =' + = (c) − + = + + = + + = Q.2 For what value of K the following system of equations possess a non trivial (i.e not all zero) solution over the set of rationals Q? x + K y + z = , x + K y − z = , x + y − z = For that value of K , find all the solutions of the system Q.3 The system of equations αx + y + z = α – x + αy + z = α – x + y + αz = α – has no solution Find α Q.4 If the equations a(y + z) = x, b(z + x) = y, c(x + y) = z have nontrivial solutions, then find the value of + + + + + Q.5 Given x = cy + bz ; y = az + cx ; z = bx + ay where x , y , z are not all zero , prove that a² + b² + c² + abc = Q.6 Given a = − ;b= − ;c= − where x, y, z are not all zero, prove that: + ab + bc + ca = ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.7 If sin q ≠ cos q and x, y, z satisfy the equations x cos p – y sin p + z = cos q + x sin p + y cos p + z = – sin q x cos(p + q) – y sin (p + q) + z = then find the value of x2 + y2 + z2 Q.8 Investigate for what values of λ, m the simultaneous equations x + y + z = 6; x + y + z = 10 & x + y + λ z = m have; (a) A unique solution (b) An infinite number of solutions (c) No solution Q.9 For what values of p , the equations : x + y + z = ; x+2y+4z = p & x + 4y + 10z = p2 have a solution? Solve them completely in each case Q.10 Solve the equations : K x + y − z = 1, x + K y − z = 2, x + y + K z = considering specially the case when K = Q.11 Let a, b, c, d are distinct numbers to be chosen from the set {1, 2, 3, 4, 5} If the least possible positive + =  can be expressed in the form where p and q ( + =  are relatively prime, then find the value of (p + q) solution for x to the system of equations Q.12 If bc + qr = ca + rp = ab + pq = − show that Q.13 If the following system of equations (a − t)x + by + cz = , bx + (c − t)y + az = and cx + ay + (b − t)z = has non−trivial solutions for different values of t , then show that we can express product of these values of t in the form of determinant Q.14 Show that the system of equations 3x – y + 4z = , x + 2y – 3z = –2 and 6x + 5y + λz = – has atleast one solution for any real number λ Find the set of solutions of λ = –5 + Q.15 Solve the system of equations ; + + + + + + + + ( ) = = = ( = )      EXERCISE–III Q.1 Q.2 If the system of equations x – Ky – z = 0, Kx – y – z = and x + y – z = has a non zero solution, then the possible values of K are (A) –1, (B) 1, (C) 0, (D) –1, [JEE 2000 (Screening)] Prove that for all values of θ, θ θ+ π ( ) (θ − π ) θ θ+ π ( ) (θ − π ) θ θ+ π ( ) π ( θ− ) =0 [ JEE 2000 (Mains), out of 100 ] Q.3 Find the real values of r for which the following system of linear equations has a non-trivial solution Also find the non-trivial solutions : rx − 2y + 3z = x + ry + 2z = 2x + rz = [ REE 2000 (Mains) , out of 100 ] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.4 Solve for x the equation + ! + ! − ! − ! =0 [ REE 2001 (Mains) , out of 100 ] Q.5 Test the consistency and solve them when consistent, the following system of equations for all values of λ x+y+z =1 x + 3y – 2z = λ 3x + (λ + 2)y – 3z = 2λ + [ REE 2001 (Mains) , out of 100 ] Q.6 Let a, b, c be real numbers with a2 + b2 + c2 = Show that the equation − − + + − + + + − + + − − + = represents a straight line Q.7 Q.8 [ JEE 2001 (Mains) , out of 100 ] The number of values of k for which the system of equations (k + 1)x + 8y = 4k kx + (k + 3)y = 3k – has infinitely many solutions is (A) (B) (C) (D) inifinite [JEE 2002 (Screening), 3] The value of λ for which the system of equations 2x – y – z = 12, x – 2y + z = –4, x + y + λz = has no solution is (A) (B) –3 (C) (D) –2 [JEE 2004 (Screening)] Q.9(a) Consider three points P = (− β − α!- − β) , Q = ( β − α!- β) and R= ( β − α + θ!β − θ! ) , where < α, β, θ < π/4 (A) P lies on the line segment RQ (B) Q lies on the line segment PR (C) R lies on the line segment QP (D) P, Q, R are non collinear (b) Consider the system of equations x – 2y + 3z = –1 – x + y – 2z = k x – 3y + 4z = STATEMENT-1 : The system of equations has no solution for k ≠ and STATEMENT-2 : The determinant − − − ≠ 0, for k ≠ (A) Statement-1 is True, Statement-2 is True ; statement-2 is a correct explanation for statement-1 (B) Statement-1 is True, Statement-2 is True ; statement-2 is NOT a correct explanation for statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True [JEE 2008, + 3] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) ANSWER KEY DETERMINANT EXERCISE–I Q.1 (b) p Q.4 −1 Q.11 Q.13 x = or x = ± (ab′ − a′b) (bc′ − b′c) (ca′ − c′a) ( + ) + Q.12 (a) x = − or x = − 2; (b) x = Q.17 λ2 ( a2 + b2 + c2 + λ) Q.19 Triangle ABC is isosceles EXERCISE–II Q.1 (b) x = , y = − , z = ; consistent (a) x = , y = , z = ; consistent (c) inconsistent Q.2 K = , x: y: z = − :1:− Q.3 –2 Q.4 Q.7 Q.8 (a) λ ≠ (b) λ = 3, m =10 (c) λ = 3, m ≠ 10 Q.9 x = + 2K , y = − 3K , z = K, when p = ; x = 2K, y = − 3K , z = K when p = ; where K ∈ R Q.10 If K≠2, +'! = + If K= 2, then x = λ, y = Q.11 19 Q.14 If λ ≠ –5 then x = ' − ! − λ = ( + + ), and z = where λ ∈ R Q.13 If λ = then x = Q.15 = ;y=– − and z = ; ;y= − and z = K where K ∈ R x = −(a + b + c) , y = ab + bc + ca , z = −abc ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) 10 EXERCISE–III r = ; x = k; y = ; z = − k where k ∈ R − {0} D Q.4 x = nπ, n ∈ I Q.5 If λ = 5, system is consistent with infinite solution given by z = K, y= Q.3 Q.1 (3K + 4) and x = – (5K + 2) where K ∈ R If λ ≠ 5, system is consistent with unique solution given by z = Q.7 B Q.8 D Q.9 (1 – λ); x = (λ + 2) and y = (a) D; (b) A ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) 11 ... vanishes then (x − a) is a factor of the determinant MULTIPLICATION OF TWO DETERMINANTS : (i) = + + + + Similarly two determinants of order three are multiplied ETOOS Academy Pvt Ltd : F-106,... can be expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants e.g + + + = + P− − 6: The value of a determinant is not altered by adding to the elements... Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) PROPERTIES OF DETERMINANTS : − : The value of a determinant remains unaltered , if the rows & columns are inter