MATRICES. Chapter Introduction. 3.2 Matrix. The essence of Mathematics lies in its freedom. CANTOR

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1 56 Chapter 3 MATRICES The essence of Mathematics lies in its freedom. CANTOR 3.1 Introduction The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our wk to a great extent when compared with other straight fward methods. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrices are not only used as a representation of the coefficients in system of linear equations, but utility of matrices far exceeds that use. Matrix notation and operations are used in electronic spreadsheet programs f personal computer, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analysing the results of an experiment etc. Also, many physical operations such as magnification, rotation and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences, but also in genetics, economics, sociology, modern psychology and industrial management. In this chapter, we shall find it interesting to become acquainted with the fundamentals of matrix and matrix algebra. 3. Matrix Suppose we wish to express the infmation that Radha has 15 notebooks. We may express it as [15] with the understanding that the number inside [ ] is the number of notebooks that Radha has. Now, if we have to express that Radha has 15 notebooks and 6 pens. We may express it as [15 6] with the understanding that first number inside [ ] is the number of notebooks while the other one is the number of pens possessed by Radha. Let us now suppose that we wish to express the infmation of possession

2 MATRICES 57 of notebooks and pens by Radha and her two friends Fauzia and Simran which is as follows: Radha has 15 notebooks and 6 pens, Fauzia has 10 notebooks and pens, Simran has 13 notebooks and 5 pens. Now this could be arranged in the tabular fm as follows: Notebooks Pens Radha 15 6 Fauzia 10 Simran 13 5 and this can be expressed as Radha Fauzia Simran Notebooks Pens 6 5 which can be expressed as: In the first arrangement the entries in the first column represent the number of note books possessed by Radha, Fauzia and Simran, respectively and the entries in the second column represent the number of pens possessed by Radha, Fauzia and Simran,

3 58 respectively. Similarly, in the second arrangement, the entries in the first row represent the number of notebooks possessed by Radha, Fauzia and Simran, respectively. The entries in the second row represent the number of pens possessed by Radha, Fauzia and Simran, respectively. An arrangement display of the above kind is called a matrix. Fmally, we define matrix as: Definition 1 A matrix is an dered rectangular array of numbers functions. The numbers functions are called the elements the entries of the matrix. We denote matrices by capital letters. The following are some examples of matrices: 5 A 0 5, i 3 B 3.5 1, x x 3 C cos x sin x + tan x In the above examples, the hizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix. Thus A has 3 rows and columns, B has 3 rows and 3 columns while C has rows and 3 columns Order of a matrix A matrix having m rows and n columns is called a matrix of der m n simply m n matrix (read as an m by n matrix). So referring to the above examples of matrices, we have A as 3 matrix, B as 3 3 matrix and C as 3 matrix. We observe that A has 3 6 elements, B and C have 9 and 6 elements, respectively. In general, an m n matrix has the following rectangular array: A [a ij ] m n, 1 i m, 1 j n i, j N Thus the i th row consists of the elements a i1, a i, a i3,..., a in, while the j th column consists of the elements a 1j, a j, a 3j,..., a mj, In general a ij, is an element lying in the i th row and j th column. We can also call it as the (i, j) th element of A. The number of elements in an m n matrix will be equal to mn.

4 MATRICES 59 Note In this chapter 1. We shall follow the notation, namely A [a ij ] m n to indicate that A is a matrix of der m n.. We shall consider only those matrices whose elements are real numbers functions taking real values. We can also represent any point (x, y) in a plane by a matrix (column row) as x y ( [x, y]). F example point P(0, 1) as a matrix representation may be given as 0 P 1 [0 1]. Observe that in this way we can also express the vertices of a closed rectilinear figure in the fm of a matrix. F example, consider a quadrilateral ABCD with vertices A (1, 0), B (3, ), C (1, 3), D ( 1, ). Now, quadrilateral ABCD in the matrix fm, can be represented as A B C D X A 1 0 B 3 Y C1 3 D 1 4 Thus, matrices can be used as representation of vertices of geometrical figures in a plane. Now, let us consider some examples. Example 1 Consider the following infmation regarding the number of men and women wkers in three facties I, II and III Men wkers Women wkers I 30 5 II 5 31 III 7 6 Represent the above infmation in the fm of a 3 matrix. What does the entry in the third row and second column represent?

5 60 Solution The infmation is represented in the fm of a 3 matrix as follows: 30 5 A The entry in the third row and second column represents the number of women wkers in facty III. Example If a matrix has 8 elements, what are the possible ders it can have? Solution We know that if a matrix is of der m n, it has mn elements. Thus, to find all possible ders of a matrix with 8 elements, we will find all dered pairs of natural numbers, whose product is 8. Thus, all possible dered pairs are (1, 8), (8, 1), (4, ), (, 4) Hence, possible ders are 1 8, 8 1, 4, 4 Example 3 Construct a 3 matrix whose elements are given by 1 aij i 3 j. Solution In general a 3 matrix is given by Now A a a a1 a a31 a 3 1 aij i 3 j, i 1,, 3 and j 1, Therefe a a a a a a 3 3 Hence the required matrix is given by A. 3 0

6 3.3 Types of Matrices In this section, we shall discuss different types of matrices. (i) Column matrix A matrix is said to be a column matrix if it has only one column. MATRICES 61 (ii) (iii) F example, 0 3 A 1 is a column matrix of der / In general, A [a ij ] m 1 is a column matrix of der m 1. Row matrix A matrix is said to be a row matrix if it has only one row. 1 F example, B 5 3 is a row matrix. 1 4 In general, B [b ij ] 1 n is a row matrix of der 1 n. Square matrix A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus an m n matrix is said to be a square matrix if m n and is known as a square matrix of der n F example A 3 1 is a square matrix of der In general, A [a ij ] m m is a square matrix of der m. Note If A [a ij ] is a square matrix of der n, then elements (entries) a 11, a,..., a nn are said to constitute the diagonal, of the matrix A. Thus, if Then the elements of the diagonal of A are 1, 4, A

7 6 (iv) (v) (vi) Diagonal matrix A square matrix B [b ij ] m m is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B [b ij ] m m is said to be a diagonal matrix if b ij 0, when i j F example, A [4], B 0, C 0 0, are diagonal matrices of der 1,, 3, respectively. Scalar matrix A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B [b ij ] n n is said to be a scalar matrix if b ij 0, when i j b ij k, when i j, f some constant k. F example A [3], B 0 1, C are scalar matrices of der 1, and 3, respectively. Identity matrix A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. In other wds, the square matrix A [a ij ] n n is an 1 if i j identity matrix, if aij. 0 if i j We denote the identity matrix of der n by I n. When der is clear from the context, we simply write it as I F example [1], 0 1, are identity matrices of der 1, and 3, respectively. Observe that a scalar matrix is an identity matrix when k 1. But every identity matrix is clearly a scalar matrix.

8 MATRICES 63 (vii) Zero matrix A matrix is said to be zero matrix null matrix if all its elements are zero. 0 0 F example, [0], 0 0, , [0, 0] are all zero matrices. We denote zero matrix by O. Its der will be clear from the context Equality of matrices Definition Two matrices A [a ij ] and B [b ij ] are said to be equal if (i) they are of the same der (ii) each element of A is equal to the cresponding element of B, that is a ij b ij f all i and j. 3 3 F example, and are equal matrices but 3 3 and are not equal matrices. Symbolically, if two matrices A and B are equal, we write A B. If x y z a 6, then x 1.5, y 0, z, a 6, b 3, c b c 3 Example 4 If x+ 3 z+ 4 y y 6 a c + b3 1 0 b Find the values of a, b, c, x, y and z. Solution As the given matrices are equal, therefe, their cresponding elements must be equal. Comparing the cresponding elements, we get x + 3 0, z + 4 6, y 7 3y a 1 3, 0 c + b 3 b + 4, Simplifying, we get a, b 7, c 1, x 3, y 5, z Example 5 Find the values of a, b, c, and d from the following equation: a+ b ab 4 3 5c d 4c+ 3d 11 4

9 64 Solution By equality of two matrices, equating the cresponding elements, we get a + b 4 5c d 11 a b 3 4c + 3d 4 Solving these equations, we get a 1, b, c 3 and d 4 EXERCISE In the matrix A 35 1, write: (i) The der of the matrix, (ii) The number of elements, (iii) Write the elements a 13, a 1, a 33, a 4, a 3.. If a matrix has 4 elements, what are the possible ders it can have? What, if it has 13 elements? 3. If a matrix has 18 elements, what are the possible ders it can have? What, if it has 5 elements? 4. Construct a matrix, A [a ij ], whose elements are given by: (i) a ij ( i+ j) i (ii) aij (iii) j 5. Construct a 3 4 matrix, whose elements are given by: 1 (i) aij 3 i+ j (ii) aij i j 6. Find the values of x, y and z from the following equations: a ij ( i+ j) 4 3 y z x+ y 6 (i) x (ii) 5+ z xy 5 8 (iii) 7. Find the value of a, b, c and d from the equation: a b a+ c 1 5 a b 3c+ d 0 13 x+ y+ z 9 x z 5 + y+ z 7

10 MATRICES A [a ij ] m n\ is a square matrix, if (A) m < n (B) m > n (C) m n (D) None of these 9. Which of the given values of x and y make the following pair of matrices equal 3x y + 1 3x, 0 y (A) x, y 7 (B) Not possible to find 3 1 (C) y 7, x (D) x, y The number of all possible matrices of der 3 3 with each entry 0 1 is: (A) 7 (B) 18 (C) 81 (D) Operations on Matrices In this section, we shall introduce certain operations on matrices, namely, addition of matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices Addition of matrices Suppose Fatima has two facties at places A and B. Each facty produces spt shoes f boys and girls in three different price categies labelled 1, and 3. The quantities produced by each facty are represented as matrices given below: Suppose Fatima wants to know the total production of spt shoes in each price categy. Then the total production In categy 1 : f boys ( ), f girls ( ) In categy : f boys ( ), f girls ( ) In categy 3 : f boys ( ), f girls ( ) This can be represented in the matrix fm as

11 66 This new matrix is the sum of the above two matrices. We observe that the sum of two matrices is a matrix obtained by adding the cresponding elements of the given matrices. Furtherme, the two matrices have to be of the same der. Thus, if A a a a b b b a1 a a3 is a 3 matrix and B b1 b b3 is another a11 + b11 a1 + b1 a13 + b13 3 matrix. Then, we define A+B a1 b1 a b a3 b In general, if A [a ij ] and B [b ij ] are two matrices of the same der, say m n. Then, the sum of the two matrices A and B is defined as a matrix C [c ij ] m n, where c ij a ij + b ij, f all possible values of i and j. Example 6 Given A 3 0 and 5 1 B 1, find A + B 3 Since A, B are of the same der 3. Therefe, addition of A and B is defined and is given by A+B Note 1. We emphasise that if A and B are not of the same der, then A + B is not defined. F example if A 1 0, B, then A + B is not defined.. We may observe that addition of matrices is an example of binary operation on the set of matrices of the same der Multiplication of a matrix by a scalar Now suppose that Fatima has doubled the production at a facty A in all categies (refer to 3.4.1).

12 Previously quantities (in standard units) produced by facty A were MATRICES 67 Revised quantities produced by facty A are as given below: Boys Girls This can be represented in the matrix fm as We observe that the new matrix is obtained by multiplying each element of the previous matrix by. In general, we may define multiplication of a matrix by a scalar as follows: if A [a ij ] m n is a matrix and k is a scalar, then ka is another matrix which is obtained by multiplying each element of A by the scalar k. In other wds, ka k [a ij ] m n [k (a ij )] m n, that is, (i, j) th element of ka is ka ij f all possible values of i and j. F example, if A , then A Negative of a matrix The negative of a matrix is denoted by A. We define A ( 1) A.

13 68 F example, let A x, then A is given by A ( 1) A ( 1) 5 x 5 x Difference of matrices If A [a ij ], B [b ij ] are two matrices of the same der, say m n, then difference A B is defined as a matrix D [d ij ], where d ij a ij b ij, f all value of i and j. In other wds, D A B A + ( 1) B, that is sum of the matrix A and the matrix B. Example 7 If Solution We have A and B , then find A B. A B Properties of matrix addition The addition of matrices satisfy the following properties: (i) Commutative Law If A [a ij ], B [b ij ] are matrices of the same der, say m n, then A + B B + A. Now A + B [a ij ] + [b ij ] [a ij + b ij ] [b ij + a ij ] (addition of numbers is commutative) ([b ij ] + [a ij ]) B + A (ii) Associative Law F any three matrices A [a ij ], B [b ij ], C [c ij ] of the same der, say m n, (A + B) + C A + (B + C). Now (A + B) + C ([a ij ] + [b ij ]) + [c ij ] [a ij + b ij ] + [c ij ] [(a ij + b ij ) + c ij ] [a ij + (b ij + c ij )] (Why?) [a ij ] + [(b ij + c ij )] [a ij ] + ([b ij ] + [c ij ]) A + (B + C)

14 (iii) (iv) MATRICES 69 Existence of additive identity Let A [a ij ] be an m n matrix and O be an m n zero matrix, then A + O O + A A. In other wds, O is the additive identity f matrix addition. The existence of additive inverse Let A [a ij ] m n be any matrix, then we have another matrix as A [ a ij ] m n such that A + ( A) ( A) + A O. So A is the additive inverse of A negative of A Properties of scalar multiplication of a matrix If A [a ij ] and B [b ij ] be two matrices of the same der, say m n, and k and l are scalars, then (i) k(a +B) k A + kb, (ii) (k + l)a k A + l A (ii) k (A + B) k ([a ij ] + [b ij ]) k [a ij + b ij ] [k (a ij + b ij )] [(k a ij ) + (k b ij )] [k a ij ] + [k b ij ] k [a ij ] + k [b ij ] ka + kb (iii) ( k + l) A (k + l) [a ij ] [(k + l) a ij ] + [k a ij ] + [l a ij ] k [a ij ] + l [a ij ] k A + l A Example 8 If A + 3X 5B. 8 0 A 4 andb 4, then find the matrix X, such that Solution We have A + 3X 5B A + 3X A 5B A A A + 3X 5B A (Matrix addition is commutative) O + 3X 5B A ( A is the additive inverse of A) 3X 5B A (O is the additive identity) X 1 3 (5B A) X

15 Example 9 Find X and Y, if 5 X+ Y 0 9 and 3 6 X Y Solution We have ( X+ Y) + ( X Y) (X + X) + (Y Y) X 0 8 X Also (X + Y) (X Y) (X X) + (Y + Y) Y Y Example 10 Find the values of x and y from the following equation: Solution We have x y x y x y

16 MATRICES 71 x y x y x and y 4 14 (Why?) x 7 3 and y 18 x 4 and y 18 i.e. x and y 9. Example 11 Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B. (i) (ii) (iii) Find the combined sales in September and October f each farmer in each variety. Find the decrease in sales from September to October. If both farmers receive % profit on gross sales, compute the profit f each farmer and f each variety sold in October. Solution (i) Combined sales in September and October f each farmer in each variety is given by

17 7 (ii) Change in sales from September to October is given by (iii) % of B B B 0.0 Thus, in October Ramkishan receives Rs 100, Rs 00 and Rs 10 as profit in the sale of each variety of rice, respectively, and Grucharan Singh receives profit of Rs 400, Rs 00 and Rs 00 in the sale of each variety of rice, respectively Multiplication of matrices Suppose Meera and Nadeem are two friends. Meera wants to buy pens and 5 sty books, while Nadeem needs 8 pens and 10 sty books. They both go to a shop to enquire about the rates which are quoted as follows: Pen Rs 5 each, sty book Rs 50 each. How much money does each need to spend? Clearly, Meera needs Rs ( ) that is Rs 60, while Nadeem needs ( ) Rs, that is Rs 540. In terms of matrix representation, we can write the above infmation as follows: Requirements Prices per piece (in Rupees) Money needed (in Rupees) Suppose that they enquire about the rates from another shop, quoted as follows: pen Rs 4 each, sty book Rs 40 each. Now, the money required by Meera and Nadeem to make purchases will be respectively Rs ( ) Rs 08 and Rs ( ) Rs 43

18 MATRICES 73 Again, the above infmation can be represented as follows: Requirements Prices per piece (in Rupees) Money needed (in Rupees) Now, the infmation in both the cases can be combined and expressed in terms of matrices as follows: Requirements Prices per piece (in Rupees) Money needed (in Rupees) The above is an example of multiplication of matrices. We observe that, f multiplication of two matrices A and B, the number of columns in A should be equal to the number of rows in B. Furtherme f getting the elements of the product matrix, we take rows of A and columns of B, multiply them element-wise and take the sum. Fmally, we define multiplication of matrices as follows: The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A [a ij ] be an m n matrix and B [b jk ] be an n p matrix. Then the product of the matrices A and B is the matrix C of der m p. To get the (i, k) th element c ik of the matrix C, we take the i th row of A and k th column of B, multiply them elementwise and take the sum of all these products. In other wds, if A [a ij ] m n, B [b jk ] n p, then the i th row of A is [a i1 a i... a in ] and the k th column of B is b b b 1k k... nk, then c ik a i1 b 1k + a i b k + a i3 b 3k a in b nk The matrix C [c ik ] m p is the product of A and B. n aij bjk. j F example, if C and D 1 1, then the product CD is defined 5 4

19 74 and is given by CD This is a matrix in which each 5 4 entry is the sum of the products across some row of C with the cresponding entries down some column of D. These four computations are Thus 13 CD Example 1 Find AB, if A and B Solution The matrix A has columns which is equal to the number of rows of B. Hence AB is defined. Now 6() + 9(7) 6(6) + 9(9) 6(0) + 9(8) AB () + 3(7) (6) + 3(9) (0) + 3(8)

20 MATRICES 75 Remark If AB is defined, then BA need not be defined. In the above example, AB is defined but BA is not defined because B has 3 column while A has only (and not 3) rows. If A, B are, respectively m n, k l matrices, then both AB and BA are defined if and only if n k and l m. In particular, if both A and B are square matrices of the same der, then both AB and BA are defined. Non-commutativity of multiplication of matrices Now, we shall see by an example that even if AB and BA are both defined, it is not necessary that AB BA. Example 13 If AB BA A and B , then find AB, BA. Show that 1 Solution Since A is a 3 matrix and B is 3 matrix. Hence AB and BA are both defined and are matrices of der and 3 3, respectively. Note that AB and BA Clearly AB BA In the above example both AB and BA are of different der and so AB BA. But one may think that perhaps AB and BA could be the same if they were of the same der. But it is not so, here we give an example to show that even if AB and BA are of same der they may not be same. Example 14 If 1 0 A 0 1 and 0 1 B 1 0, then 0 1 AB and BA 1 0. Clearly AB BA. Thus matrix multiplication is not commutative.

21 76 Note This does not mean that AB BA f every pair of matrices A, B f which AB and BA, are defined. F instance, If A, B 0 0 4, then AB BA Observe that multiplication of diagonal matrices of same der will be commutative. Zero matrix as the product of two non zero matrices We know that, f real numbers a, b if ab 0, then either a 0 b 0. This need not be true f matrices, we will observe this through an example. Example 15 Find AB, if 0 1 A 0 and 3 5 B Solution We have AB Thus, if the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix Properties of multiplication of matrices The multiplication of matrices possesses the following properties, which we state without proof. 1. The associative law F any three matrices A, B and C. We have (AB) C A (BC), whenever both sides of the equality are defined.. The distributive law F three matrices A, B and C. (i) A (B+C) AB + AC (ii) (A+B) C AC + BC, whenever both sides of equality are defined. 3. The existence of multiplicative identity F every square matrix A, there exist an identity matrix of same der such that IA AI A. Now, we shall verify these properties by examples. Example 16 If A 0 3, B 0 and C 0 1, find A(BC), (AB)C and show that (AB)C A(BC).

22 MATRICES 77 Solution We have AB (AB) (C) Now BC Therefe A(BC) Clearly, (AB) C A (BC)

23 Example 17 If A 6 0 8,B 1 0,C Calculate AC, BC and (A + B)C. Also, verify that (A + B)C AC + BC Solution Now, A+B So (A + B) C Further AC and BC So AC + BC Clearly, (A + B) C AC + BC 1 3 Example 18 If A 3 1, then show that A 3 3A 40 I O Solution We have A A.A

24 MATRICES 79 So A 3 A A Now A 3 3A 40I O Example 19 In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise) is given in matrix A as A Cost per contact 40 Telephone 100 Housecall 50 Letter The number of contacts of each type made in two cities X and Y is given by Telephone Housecall Letter X B ,000. Find the total amount spent by the group in the two Y cities X and Y.

25 80 Solution We have BA 40, , ,000 X 10, , ,000 Y 340,000 X 70,000 Y So the total amount spent by the group in the two cities is 340,000 paise and 70,000 paise, i.e., Rs 3400 and Rs 700, respectively. EXERCISE Let A,B,C Find each of the following: (i) A + B (ii) A B (iii) 3A C (iv) AB (v) BA. Compute the following: (i) a b a b a + b b + c ab bc + b a b a (ii) + a c a b ac ab + + (iii) cos x sin x sin x cos x + (iv) sin x cos x cos x sin x 3. Compute the indicated products. (i) a b a b b a b a (ii) 1 [ 3 4] (iii) (iv) (v) (vi)

26 MATRICES If A 5 0,B 4 5 andc 0 3, then compute (A+B) and (B C). Also, verify that A + (B C) (A + B) C. 5. If A and B , then compute 3A 5B cosθ sin θ sin θ cosθ Simplify cos θ + sinθ sin θ cosθ cosθ sin θ 7. Find X and Y, if (i) X+Y and X Y (ii) 3 X + 3Y and 3X Y Find X, if Y 9. Find x and y, if and X + Y y x Solve the equation f x, y, z and t, if x z y t If 1 10 x y , find the values of x and y. 1. Given x y x 6 4 x+ y 3 z w + 1 w z+ w 3, find the values of x, y, z and w.

27 8 cos x sin x If F( x) sinx cosx 0, show that F(x) F(y) F(x + y) Show that (i) (ii) Find A 5A + 6I, if 0 1 A If 17. If 1 0 A 0 1, prove that A 3 6A + 7A + I A andi 4 0 1, find k so that A ka I 18. If α 0 tan A α tan 0 and I is the identity matrix of der, show that cosα sin α I + A (I A) sin α cosα 19. A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (a) Rs 1800 (b) Rs 000

28 MATRICES The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra. Assume X, Y, Z, W and P are matrices of der n, 3 k, p, n 3 and p k, respectively. Choose the crect answer in Exercises 1 and. 1. The restriction on n, k and p so that PY + WY will be defined are: (A) k 3, p n (B) k is arbitrary, p (C) p is arbitrary, k 3 (D) k, p 3. If n p, then the der of the matrix 7X 5Z is: (A) p (B) n (C) n 3 (D) p n 3.5. Transpose of a Matrix In this section, we shall learn about transpose of a matrix and special types of matrices such as symmetric and skew symmetric matrices. Definition 3 If A [a ij ] be an m n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A (A T ). In other wds, if A [a ij ] m n, then A [a ji ] n m. F example, if A 3 1, then A Properties of transpose of the matrices We now state the following properties of transpose of matrices without proof. These may be verified by taking suitable examples. F any matrices A and B of suitable ders, we have (i) (A ) A, (ii) (ka) ka (where k is any constant) (iii) (A + B) A + B (iv) (A B) B A Example 0 If A and B , verify that (i) (A ) A, (ii) (A + B) A + B, (iii) (kb) kb, where k is any constant.

29 84 Solution (i) We have (ii) A ( ) 3 3 A 3 A A Thus (A ) A We have 3 3 A, B A B Therefe (A + B) Now A ,B 1, 0 4 (iii) So 5 5 A + B Thus (A + B) A + B We have kb k 1 k k k 1 4 k k 4k Then (kb) k k 1 k k k 1 kb k 4k 4 Thus (kb) kb

30 MATRICES 85 5 Example 1 If A 4,B[ 1 3 6] Solution We have A 4,B[ 1 3 6] 5, verify that (AB) B A. 5 then AB 4 [ 1 3 6] Now A [ 4 5], 1 B 3 6 Clearly (AB) B A [ ] (AB) B A 3.6 Symmetric and Skew Symmetric Matrices Definition 4 A square matrix A [a ij ] is said to be symmetric if A A, that is, [a ij ] [a ji ] f all possible values of i and j. F example 3 3 A is a symmetric matrix as A A Definition 5 A square matrix A [a ij ] is said to be skew symmetric matrix if A A, that is a ji a ij f all possible values of i and j. Now, if we put i j, we have a ii a ii. Therefe a ii 0 a ii 0 f all i s. This means that all the diagonal elements of a skew symmetric matrix are zero.

31 86 0 e F example, the matrix B e 0 f g f g is a skew symmetric matrix as B B 0 Now, we are going to prove some results of symmetric and skew-symmetric matrices. Theem 1 F any square matrix A with real number entries, A + A is a symmetric matrix and A A is a skew symmetric matrix. Proof Let B A + A, then B (A + A ) A + (A ) (as (A + B) A + B ) A + A (as (A ) A) A + A (as A + B B + A) B Therefe B A + A is a symmetric matrix Now let C A A C (A A ) A (A ) (Why?) A A (Why?) (A A ) C Therefe C A A is a skew symmetric matrix. Theem Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix. Proof Let A be a square matrix, then we can write 1 1 A (A+ A) + (AA ) From the Theem 1, we know that (A + A ) is a symmetric matrix and (A A ) is a skew symmetric matrix. Since f any matrix A, (ka) ka, it follows that 1 (A + A ) is symmetric matrix and 1 (A A ) is skew symmetric matrix. Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

32 MATRICES 87 4 Example Express the matrix B as the sum of a symmetric and a 1 3 skew symmetric matrix. Solution Here B Let P (B + B ) Now P P , 1 3 Thus P 1 (B + B ) is a symmetric matrix. Also, let Q (B B ) Then Q 0 3 Q 5 3 0

33 88 Thus Q 1 (B B ) is a skew symmetric matrix. Now P+Q B Thus, B is represented as the sum of a symmetric and a skew symmetric matrix. EXERCISE Find the transpose of each of the following matrices: (i) (ii) 3 (iii) If A and B 1 0, then verify that (i) (A + B) A + B, (ii) (A B) A B If A 1 and B 1 3, then verify that 0 1 (i) (A + B) A + B (ii) (A B) A B If A and B 1 1, then find (A + B) 5. F the matrices A and B, verify that (AB) B A, where 1 3 (i) A 4, B [ 1 1] 0 (ii) A 1, B [ 1 5 7]

34 MATRICES If (i) cosα sin α A sin α cosα, then verify that A A I (ii) If sin α cos α A cosα sin α, then verify that A A I 7. (i) Show that the matrix A is a symmetric matrix. (ii) Show that the matrix A is a skew symmetric matrix F the matrix A 6 7, verify that (i) (A + A ) is a symmetric matrix (ii) (A A ) is a skew symmetric matrix 1 9. Find ( A A ) 1 + and ( A A ), when 0 a A a 0 b c b c Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) (ii) (iii) (iv) 1 5 1

35 90 Choose the crect answer in the Exercises 11 and If A, B are symmetric matrices of same der, then AB BA is a (A) Skew symmetric matrix (B) Symmetric matrix (C) Zero matrix (D) Identity matrix 1. If (A) cosα sin α A, sin α cosα then A + A I, if the value of α is π 6 (C) π (B) (D) 3.7 Elementary Operation (Transfmation) of a Matrix There are six operations (transfmations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations transfmations. (i) (ii) (iii) The interchange of any two rows two columns. Symbolically the interchange of i th and j th rows is denoted by R i R j and interchange of i th and j th column is denoted by C i C j. F example, applying R 1 R to π 3 3π 1 1 A 1 3 1, we get The multiplication of the elements of any row column by a non zero number. Symbolically, the multiplication of each element of the i th row by k, where k 0 is denoted by R i k R i. The cresponding column operation is denoted by C i kc i F example, applying C3 C, tob 3, we get The addition to the elements of any row column, the cresponding elements of any other row column multiplied by any non zero number. Symbolically, the addition to the elements of i th row, the cresponding elements of j th row multiplied by k is denoted by R i R i + kr j

36 The cresponding column operation is denoted by C i C i + kc j. F example, applying R R R 1, to MATRICES 91 1 C 1, we get Invertible Matrices Definition 6 If A is a square matrix of der m, and if there exists another square matrix B of the same der m, such that AB BA I, then B is called the inverse matrix of A and it is denoted by A 1. In that case A is said to be invertible. F example, let A 3 1 and B 3 1 be two matrices. Now AB I Also BA 1 0 I 0 1. Thus B is the inverse of A, in other wds B A 1 and A is inverse of B, i.e., A B 1 Note 1. A rectangular matrix does not possess inverse matrix, since f products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same der.. If B is the inverse of A, then A is also the inverse of B. Theem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. Proof Let A [a ij ] be a square matrix of der m. If possible, let B and C be two inverses of A. We shall show that B C. Since B is the inverse of A AB BA I... (1) Since C is also the inverse of A AC CA I... () Thus B BI B (AC) (BA) C IC C Theem 4 If A and B are invertible matrices of the same der, then (AB) 1 B 1 A 1.

37 9 Proof From the definition of inverse of a matrix, we have (AB) (AB) 1 1 A 1 (AB) (AB) 1 A 1 I (Pre multiplying both sides by A 1 ) (A 1 A) B (AB) 1 A 1 (Since A 1 I A 1 ) IB (AB) 1 A 1 B (AB) 1 A 1 B 1 B (AB) 1 B 1 A 1 I (AB) 1 B 1 A 1 Hence (AB) 1 B 1 A Inverse of a matrix by elementary operations Let X, A and B be matrices of, the same der such that X AB. In der to apply a sequence of elementary row operations on the matrix equation X AB, we will apply these row operations simultaneously on X and on the first matrix A of the product AB on RHS. Similarly, in der to apply a sequence of elementary column operations on the matrix equation X AB, we will apply, these operations simultaneously on X and on the second matrix B of the product AB on RHS. In view of the above discussion, we conclude that if A is a matrix such that A 1 exists, then to find A 1 using elementary row operations, write A IA and apply a sequence of row operation on A IA till we get, I BA. The matrix B will be the inverse of A. Similarly, if we wish to find A 1 using column operations, then, write A AI and apply a sequence of column operations on A AI till we get, I AB. Remark In case, after applying one me elementary row (column) operations on A IA (A AI), if we obtain all zeros in one me rows of the matrix A on L.H.S., then A 1 does not exist. Example 3 By using elementary operations, find the inverse of the matrix 1 A 1. Solution In der to use elementary row operations we may write A IA A, then A (applying R R R 1 )

38 MATRICES A (applying R 1 5 R ) A (applying R 1 1 R 1 R ) Thus A Alternatively, in der to use elementary column operations, we write A AI, i.e., A 0 1 Applying C C C 1, we get A Now applying C C, we have A Finally, applying C 1 C 1 C, we obtain Hence A A

39 94 Example 4 Obtain the inverse of the following matrix using elementary operations 0 1 A Solution Write A I A, i.e., A A (applying R 1 R ) A (applying R 3 R 3 3R 1 ) A (applying R 1 R 1 R ) A (applying R 3 R 3 + 5R ) A (applying R 3 1 R 3 ) A (applying R 1 R 1 + R 3 ) 5 3 1

40 MATRICES Hence A 1 Alternatively, write A AI, i.e., A (applying R R R 3 ) A A (C 1 C ) A (C 3 C 3 C 1 ) A (C 3 C 3 + C ) A (C 3 1 C 3 )

41 A (C 1 C 1 C ) A (C 1 C 1 + 5C 3 ) A4 3 1 (C C 3C 3 ) 5 3 Hence A Example 5 Find P 1, if it exists, given 10 P 5 1. Solution We have P I P, i.e., P P 0 1 (applying R R 1 )

42 MATRICES P (applying R R + 5R 1 ) 1 We have all zeros in the second row of the left hand side matrix of the above equation. Therefe, P 1 does not exist. EXERCISE 3.4 Using elementary transfmations, find the inverse of each of the matrices, if it exists in Exercises 1 to Matrices A and B will be inverse of each other only if (A) AB BA (B) AB BA 0 (C) AB 0, BA I (D) AB BA I

43 98 Miscellaneous Examples Example 6 If cosθ sin θ A sin θ cosθ, then prove that n cosnθ sin nθ A sin nθ cosnθ, n N. Solution We shall prove the result by using principle of mathematical induction. cosθ sin θ We have P(n) : If A sin θ cosθ, then n cosnθ sin nθ A sin nθ cosnθ, n N cosθ sin θ P(1) : A sin θ cosθ, so Therefe, the result is true f n 1. Let the result be true f n k. So 1 cosθ sin θ A sin θ cosθ cosθ sin θ P(k) : A sin θ cosθ, then Now, we prove that the result holds f n k +1 k cos kθ sin kθ A sin kθ coskθ Now A k + 1 k cosθ sin θ coskθ sin kθ AA sin θ cosθ sin kθ coskθ cosθcos kθ sin θsin kθ cosθsin kθ+ sinθcos kθ sin θcos kθ+ cosθsin kθ sinθsin kθ+ cosθcoskθ cos( θ+ kθ) sin ( θ+ kθ ) cos( k + 1) θ sin ( k + 1) θ sin ( θ+ kθ) cos( θ+ kθ) sin ( k + 1) θ cos( k + 1) θ Therefe, the result is true f n k + 1. Thus by principle of mathematical induction, n cos nθ sin nθ we have A sin nθ cosnθ, holds f all natural numbers. Example 7 If A and B are symmetric matrices of the same der, then show that AB is symmetric if and only if A and B commute, that is AB BA. Solution Since A and B are both symmetric matrices, therefe A A and B B. Let AB be symmetric, then (AB) AB

44 But (AB) B A BA (Why?) Therefe BA AB Conversely, if AB BA, then we shall show that AB is symmetric. Now (AB) B A B A (as A and B are symmetric) AB Hence AB is symmetric. MATRICES Example 8 Let A,B,C Find a matrix D such that CD AB O. Solution Since A, B, C are all square matrices of der, and CD AB is well defined, D must be a square matrix of der. a b Let D c d. Then CD AB 0 gives 5 a b c d O a+ 5c b+ 5d 3 0 3a+ 8c 3b+ 8d a+ 5c 3 b+ 5d 3a+ 8c 43 3b+ 8d By equality of matrices, we get a + 5c (1) 3a + 8c () b + 5d 0... (3) and 3b + 8d 0... (4) Solving (1) and (), we get a 191, c 77. Solving (3) and (4), we get b 110, d 44. Therefe a b D c d 77 44

45 100 Miscellaneous Exercise on Chapter Let A 0 0, show that (ai + ba)n a n I + na n 1 ba, where I is the identity matrix of der and n N.. If A 1 1 1, prove that n1 n1 n n n1 n1 n1 A 3 3 3, n N. n1 n1 n n 1+ n 4n 3. If A, then prove that A 1 1 n 1n, where n is any positive integer. 4. If A and B are symmetric matrices, prove that AB BA is a skew symmetric matrix. 5. Show that the matrix B AB is symmetric skew symmetric accding as A is symmetric skew symmetric. 6. Find the values of x, y, z if the matrix A A I. 0 y z A x y z x y z satisfy the equation 7. F what values of x : [ ] x O? 8. If 3 1 A 1, show that A 5A + 7I Find x, if [ x ] 1 0 x O 0 3 1

46 MATRICES A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below: Market Products I 10,000,000 18,000 II 6,000 0,000 8,000 (a) If unit sale prices of x, y and z are Rs.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra. (b) If the unit costs of the above three commodities are Rs.00, Rs 1.00 and 50 paise respectively. Find the gross profit Find the matrix X so that X If A and B are square matrices of the same der such that AB BA, then prove by induction that AB n B n A. Further, prove that (AB) n A n B n f all n N. Choose the crect answer in the following questions: 13. If A α β γ α is such that A² I, then (A) 1 + α² + βγ 0 (B) 1 α² + βγ 0 (C) 1 α² βγ 0 (D) 1 + α² βγ If the matrix A is both symmetric and skew symmetric, then (A) A is a diagonal matrix (B) A is a zero matrix (C) A is a square matrix (D) None of these 15. If A is square matrix such that A A, then (I + A)³ 7 A is equal to (A) A (B) I A (C) I (D) 3A Summary A matrix is an dered rectangular array of numbers functions. A matrix having m rows and n columns is called a matrix of der m n. [a ij ] m 1 is a column matrix. [a ij ] 1 n is a row matrix. An m n matrix is a square matrix if m n. A [a ij ] m m is a diagonal matrix if a ij 0, when i j.

47 10 A [a ij ] n n is a scalar matrix if a ij 0, when i j, a ij k, (k is some constant), when i j. A [a ij ] n n is an identity matrix, if a ij 1, when i j, a ij 0, when i j. A zero matrix has all its elements as zero. A [a ij ] [b ] B if (i) A and B are of same der, (ii) a b f all ij ij ij possible values of i and j. ka k[a ij ] m n [k(a ij )] m n A ( 1)A A B A + ( 1) B A + B B + A (A + B) + C A + (B + C), where A, B and C are of same der. k(a + B) ka + kb, where A and B are of same der, k is constant. (k + l ) A ka + la, where k and l are constant. If A [a ij ] m n and B [b jk ] n p, then AB C [c ik ] m p, where c n a b ik ij jk j 1 (i) A(BC) (AB)C, (ii) A(B + C) AB + AC, (iii) (A + B)C AC + BC If A [a ij ] m n, then A A T [a ji ] n m (i) (A ) A, (ii) (ka) ka, (iii) (A + B) A + B, (iv) (AB) B A A is a symmetric matrix if A A. A is a skew symmetric matrix if A A. Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix. Elementary operations of a matrix are as follows: (i) R i R j C i C j (ii) R i kr i C i kc i (iii) R i R i + kr j C i C i + kc j If A and B are two square matrices such that AB BA I, then B is the inverse matrix of A and is denoted by A 1 and A is the inverse of B. Inverse of a square matrix, if it exists, is unique.