Matrices and Determinants Teaching-Learning Points A matri is an ordered rectanguar arra (arrangement) of numbers and encosed b capita bracket [ ]. These numbers are caed eements of the matri. Matri is denoted b capita etters of the Engish aphabet and its eements are denoted b sma etters. n a matri horizonta ines of numbers are caed rows of the matri and vertica ines are caed coumns of the matri. A matri having m rows and n coumns is caed a matri of order m b n, written as m n. n genera a matri of order m n is written as: (OR) A = Fa a a a j a a a a a j a a a a a j a a a a a j a J a a a a j a n n n i i i i in m m m m mn A = [a ij where L i L m, L J L n, m, n M N Tpes of matrices : A matri A = [a ij is said to be a: (i) Row mati if m = (ii) Coumn matri if n = (iii) Zero/Nu matri if each of its eements is zero. (iv) Square matri if m = n (v) Diagona matri, if m = n and a ij = 0 when i N J (vi) Scaar matri, if m = n, a ij = 0 when i N j and a ij = k when i = J (vii) Unit/identit matri if m = n, a ij = 0 when i ¹ j G K Two matrices A = [a ij ] p q are said to be equa (i.e. A = B) if m = p, n = q and a ij = b ij for a i, j. Scaar mutipication of a matri. Let A = [a ij and k be an scaar, then KA = K[a ij = [Ka ij. i.e. to mutip a matri b a scaar mutip each eement of the matri b the scaar.
Addition of matrices: Let A = [a ij, then A + B = [a ij + b ij Properties of matri addition. (i) A + B = B + A (commutative aw) (ii) (A + B) + C = A + (B + C) [Associative aw] (iii)a + O = A = O + A, where O is Nue matri (eistence of additive identit) (iv)a + ( A) = O = ( A) + A [eistence of additive inverse] A B = A + ( B). Mutipication of matrices. Let A = [a ij ] p q be two matrices, then the product of A and B (i.e. AB) is defined if n = p and it is a matri of order m q. Let A = [a ij ] n p, then A B = C (sa) = [c ij ] m b, where c ij is obtained b taking ith row of A and jth coumn of B, mutiping their corresponding eements and taking sum of these produce. (i) n genera, AB O BA. (ii) (AB)C = A(BC) (iii) A. = A where is identit matri. (iv) A(B + C) = AB + AC (or) (A + B) C = AC + BC. (Distributive Law) Transpose of a matri A is obtained b interchanging its rows and coumns. t is denoted b A. Properties of transpose: (i) (KA) = KA (ii) (A + B) = A + B (iii) AB) = B A (iv) (A ) = A. A square matri is caed, a smmetric matri if A = A and a skew smmetric matri if A = A. For an square matri A, A + A is awas smmetric matri and A A is awas skew smmetric matri. Ever square matri can be epressed as a sum of smmetric and skew smmetric matri. P Q P Q A R A A A i.e. A = S R U TV SU TV Eementar operations (transformations) of a matri. (i) nterchange of an two rows (or two coumns) i.e. R i W R j W C j (ii) Mutipication of the eements of an row (or an coumn) b a non zero number i.e. i.e. R i X KR i X KC i (iii) Addition of the eements of an row (or an coumn) to the corresponding eements of an other row (or coumn) mutipied b an non zero number i.e. i.e. R i X R i + KR j = C i + KC i
A square matri A is said to be invertibe if there eists another square matri B of the same order such that AB = = BA, then B is caed the inverse of A and is denoted b A. Properties of inverse of matri. (i) AB = Y B = A and A = B (ii) (A ) = A (iii) (AB) = B A (iv) (A ) = (A ) For finding the inverse of a square matri b using. (i) eementar row operations (transformations) we write A = A. (ii) eementar coumn operations (transformations) we write A = A. A number which is associated to a square matri A = [a ij ] n n is caed a determinant of the matri A and it is denoted as A. Determinant ca be epanded b using an row or coumn. Properties of determinants: (i) A = A. (ii) R i Z R j Z C j Y A = A (iii) f R i is identica to R j or C i is identica to C j Y A = 0. (iv) ka kb kc = k = k A n genera k A n [ n = k n A \ n n (v) R i ] R i + kr j = C i + kc j Y A remains unchanged (vi) f some or a eements of a row (or coumn) of a determinant are epressed as the sum two or more terms, then the determinant can be epressed as a sum of two or more determinants. AB = A B A = A A square matri is said to be singuar if A = 0 and non singuar if A ^ 0 Using determinants Area if a _ABC with vertices A( ), B( ), C( ) is given b Minor of an eement a ij of a determinants is the determinant obtained b deeting the ith row and th coumn in which eements a ij ies. t is denoted b M ij. Cofactor of an eement a ij of a determinant is denoted as A ij and is defined as A ij = ( ) i + M ij Adjoint of a square matri A = [a ij ] n n is defined as the transpose of the matri [A ij ] n n where A ij is the cofactor of the eement a ij and it is denoted as adj.a 4 4
A(adjA) = (adja) A = A adja = A n, where A is a square matri of order n. A square matri A is invertibe if and on A ` 0 if A is a non singuar matri adja A =. A f A and B are non singuar matrices of the same order then AB and BA are aso non singuar matrices of the same order. A sstem of inear equations in three variabes can be epressed in a matri equation. For eampe, a + b + c z = d a + b + c z = d a + b + c z = d a b c bc d e d e d e d e d e dz e f g f g = hd i j d k j k j d k m a AX = B where A = n o p q p q, X = pr q s tu v w v w, B = v z w z d { d } } ~ d } B soving the matri equation AX = B or X = A B we get soution of given sstem of equations. f A ` 0, then the sstem of equations has unique soution and hence it is consistent. f A = 0 and (adja) B ` 0 (zero matri), then the sstem of equations has no soution and hence it is inconsistent. f A = 0 and (adja) B = 0 (zero matri), then the sstem of equations ma be either consistent or inconsistent according as the sstem has either infinite man soutions or no soution. Question for Practice Ver short answer questions carring one mark each:. construct a matri A = [a ij ], where a J i J i. f ƒ a b 5 a ˆ = 6 Š 5 8 Œ, find the vaues of a and b. Ž. f a b = 0 5, find a and b. š 4. f a matri has eements, then how man possibe orders it can have.
œcos sin 5. f A = ž sin cos Ÿ, 0 < < and A + A =, then find the vaue of. 4 6. f = P + where P is smmetric and is skew smmetric matri, find. cos ª sin ª 7. Evauate sin ª cosª 8. Find the vaue of if 4 «9. What is the vaue of the determinant 0. Find cofactor of a in 0 4 5 0. Find the minor of a in the determinant 5 6 0 5 4. Let A be a square matri of order and A = 5, find the vaue of A.. Let A be a square matri of order and A =, then find adja. 4. f A is invertibe matri of order and A =, then find A. 5. f a singuar matri A is given b 4, then find the vaue of. ± Short answer questions carring 4 marks each: 6. Epress the foowing matri as a sum of a smmetric and a skew smmetric matri. ² 5³ 6 8 µ µ 4 6 5µ 7. Given that A = ¹ º 4» ¼ ½ and = ¾ 0 À 0 Á Â Ã, find the rea number k such that A ka + = 0. Ä Å 8. Given that A = Æ Ç Ê Æ Ç and B = Ë Ì 0 4 5 Í ÆÈ 4 0Ç Î Ï, verif that (AB) = B A. É 9. f A = Ð ¹ º 4» ¼ ½ show that A 6A + 7 = 0. ence find A.
0. Show that = is one of the roots of the equation. Using properties of determinants show that: 6 Ñ = 0. Find other roots aso. a b c a b c = (a b) (b c) (c a). b Ò c c Ò a a Ò b q Ò r r Ò p p Ò q Ò z z Ò Ò = p q r z. z z z z z = ( + + z) 4. a b c c Ñ b a Ñ c b c a a b a Ñ b c = (a + b + c) (a + b + c ) 5. = ( ). Long Answer tpe questions carring 6 marks each: 6. Ó Ô Find the inverse of the matri using eementar transformations: Õ Ö Õ Ö Õ ÖØ 7. Sove the foowing sstem of inear equations using inverse of matri. + + 4z = 8, + z =, 4 5z = 9. 8. f A = Ù 5Ú Û 4 Ü, find A Û Ü. ence sove the foowing sstem of inear equations: ÛÝ ÜÞ + 5z = + 4z = 5 + z =
ß à 9. f A = á â, find A á â and hence sove the foowing sstem of inear equations: áã 5 âä + + 5z = 0 [int : (A ) = (A ) z = + z = å æ ë 0 ì 0. f A = ç 0 è and B = í 9 î. Find AB and hence sove the foowing sstem of equation. ç è í î çé 4èê íï 6 îñ + z = [int : AB = ò A = B] z = + 4z = Answers. ó ô õ 5 4 ö ø. a = 8, b =. a =, b = 4 4. 6 ù ú 0 û 5. 6. ü 0 ý 7. 8. þ ÿ 9. 0 0.. 0. 40. 4 4. 5. 6. ð ð 0 9 9 8 9 9 0 9 5 9 0 0.,. 8. 9. A = 4 7 6. 5 0 5 5 5 0 5 5 5 7. =, =, z =. 8. 0 9 5 =, =, z =. 9. A = ß 4 5à á 7 â =, =, z =. 0. = 0, = 5, z =. 7 á â áã 6 âä