MATRICES

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1 MARICES 1. Matrix: he arrangement of numbers or letters in the horizontal and vertical lines so that each horizontal line contains same number of elements and each vertical row contains the same numbers of elements Ex: a b c. Horizontal lines are called rows. Vertical lines are called columns. 3. Order of a matrix: he number of rows and number of columns of a matrix is called order of matrix. If a matrix contains m rows and n columns then its order is denoted by m x n {It is read as a m by n} 4. Generally matrix of order m x n is denoted by ( aij ) m n 5. Row matrix: If a matrix contains only one row then the matrix is called row matrix 1 3 general form of representing a row matrix is ( aij ) 1 n Ex: [ ] Column matrix: If a matrix contains only one column then the matrix is called column matrix Ex: a b c 31 General form of representing a column matrix is ( aij ) m 1 7. Rectangular matrix: If the number of rows of a matrix in not equal to the number of columns of the matrix the matrix then the matrix is called rectangular matrix Ex : a b c p q r 3

2 8. Square matrix: If the number of rows of a matrix is equal to the number of columns of the matrix then the matrix is called a square matrix. Ex: a c b d a h g h b f g f c 9. Principal diagonal : In a square matrix the diagonal joining the first row first column to the last row last column is called principal diagonal (or) leading diagonal the principal diagonal is as shown below. a c b d a1 b1 c1 a b c a 3 b3 c race of a matrix: he sum of the principal diagonal elements of a square matrix is called trace of a matrix If A ( aij ) n n is a square matrix then a11 + a + a ann i.e trace of matrix A denoted by tr (A) n aij is called i Lower triangular matrix: A square matrix ( aij ) n n matrix when aij 0 i< j is said to be a lower triangular Ex : a a1 a 0 a 33 a3 a Upper triangular matrix: A square matrix ( aij ) n n is said to be upper triangular matrix when aij 0 i > j Eg : riangular matrix: A square matrix is said to be a triangular matrix if it is either upper triangular matrix or lower triangular matrix

3 14. Diagonal matrix: A square matrix is said to be a diagonal matrix if all the principal diagonal elements are non zero and all the remaining elements are zero A square matrix ( aij i j ) n n is said to be diagonal matrix when aij 0 i j, 0 Ex: he matrix a b c is also denoted by diag {a, b, c} 15. Scalar matrix: A square matrix aij k ( 0) when i j 0 when i j Eg : Unit matrix: A square matrix ( aij ) n n is said to be unit matrix when aij 1 when i j 0 when i j Eg : Generally unit matrix is denoted by I 17. Null Matrix: A matrix is said to be a null matrix if all the elements are zeros. 18. Equality of matrices: wo matrices A and B are said to be equal if i) A and B are of same order and ii) he corresponding elements of A and B are the same

4 if A a a a a1 a a 3 and B b b b b1 b b 3 are equal then a 11 b11 a1 b a b a 1 b1 a b 3 3 a b 19. Definition: {sum of two matrices}: Let A and B be matrices of same order. hen the sum of A and B denoted by A + B is defined as the matrix of the same order in which each element is the sum of corresponding elements of A and B If A ( aij ) m n ; B ( bij ) m n then A+ B ( cij ) m n cij aij + bij 0. (i) matrix addition is commutative i.e. A + B B + A ii) Matrix addition is associative i.e. A + (B + C) (A + B) + C iii) If A is m x n matrix and O is the m x n null matrix A + O O +A A O is called the additive identify iv) If A is m x n matrix there is unique m x n matrix B such that A + B B + A O, O being the m x n null matrix. his B is denoted by A and is called the additive inverse of A. 1. Scalar multiplication of matrix: Let A be a matrix of order m x n and k be a scalar then the m x n matrix obtained by multiplying each element of A by k is called a scalar multiple of A and is denoted by k A. Properties of scalar multiplication: Let A and B be matrices of the same order and α, β be scalar then i) α( β A) ( αβ) A β( α A) ii) ( α + β )A α A+ β A iii) OA O iv) α( A+ β ) αa+ αβ

5 1. Multiplication of Matrices: wo matrices A and B are confirmable for multiplication when the number of columns of A is equal to the number of rows of B.. Definition: (Product of two matrices) Let A ( aij ) m n and B ( bij ) n p where n cij aik bkj is called product of A and B and is denoted by AB. k 1 i) If A, B are two matrices such that AB exist then BA need not exist ii) If AB and BA both exist then they need not be equal iii) Matrix product is not commutative Properties of Multiplication: 1) If A, B, C are three matrices then A (BC) (AB) C ) i.e. matrix product is associative i) Matrix product is distributive over addition i.e. A (B + C) AB + AC ii) If A, B are two matrices such that AB 0 then it is not necessary that either A 0 or B 0 or both A and B are null matrices iii) If A, B, C are three matrices such that AB AC then it is not necessary that either a 0 or B C iv) If A is a square matrix of order n and I is an identify matrix of order n then AI IA A 3. Idem potent matrix: A square matrix A is said to be an idempotent matrix if A A 4. Involutory matrix: A square matrix A is said to be involutory matrix if A I 5. Nill Potent matrix: A square matrix A is said to be nill point matrix when n A 0 the least value of n is called the index of nill point matrix 6. ranspose of a matrix: If A ( aij ) m n is a matrix then the matrix obtained by interchanging the rows into columns is called the transpose of A this is denoted by A or A If A ( aij ) m n then A ( aij ) n m

6 7. Properties of ranspose: i) ( A ) A ii) ( A + B) A + B iii) ( A B) A B iv) ( AB) B A v) ( KA) KA 8. Symmetric matrix: A square matrix A is said to be symmetric matrix if A A 9. Skew symmetric matrix: A square matrix A is said to be skew symmetric matrix if A A 10. Every square matrix can be uniquely expressed as the sum of symmetric and skew symmetric matrices 1 1 A A+ A + A A 1 matrix. { } { } Where { A+ A } is a symmetric matrix and { A A } 3.4 Determinants 1. Determinant of x matrix 1 is skew symmetric Def : If a c b d is a x matrix then ad ac is called determinant of a matrix. Minor of an element: he determinant of a square matrix obtained by eliminating the row and column in which the element is present. his is denoted by Mij. 3. Cofactor of an element: he cofactor of an element in the i th row and its minor multidied by ( 1) i+ j generally this is denoted by Aij th j defined as 4. Definition {determinant of 3 x 3 matrix}: he sum of the products of the elements of a row or column with their cofactor is called determinant of the matrix

7 Singular matrix : If the determinant of a matrix is zero then the matrix is called singular matrix Non-singular matrix : If the determinant of a matrix is non zero then the matrix is called non- singular matrix. Properties of Determinant : 1. he sum of the products of the element of a row or column of a square matrix with their corresponding cofactors is the determinant of matrix this is denoted by A. he determinant of a matrix A is same as the determinant of its transpose i.e. A A 3. If two rows or columns of a square matrix are interchanged then the determinant changes its sign. 4. It two rows or column of a square matrix are identical then the value of the determinant is zero 5. If all the elements of a row or column of a square matrix are multiplied by a constant k then the determinant is also multiplied by the same constant k 6. If A is a square matrix of order n then n KA K A 7. If all the elements of a row or column of a square matrix are k times the elements of any row or column then the value of the determinant is zero 8. If all the elements of a row are expressed as the sum of two elements then the determinant can also be expressed as the sum of two determinants. 9. If all the elements of a row or column of a square matrix are added to k times of the corresponding elements of any other row or column then the value of the determinant remains un altered 10. Sum of the product of the element of row with the corresponding cofactor of any other row or column is zero 11. If A, B are two square matrices of same order then AB A B 1. If all the elements of a determinant of a square are the polynomials of x and by wrify x a if two rows are identical then x a is a factor for the determinant n n 13. If A is a square matrix then det ( A ) ( det A).

8 1. Adjoint of a matrix: - he transpose of a matrix obtained by replacing the elements of a matrix with their corresponding cofactors is called adjoint of the matrix,. If A is a matrix then its adjoint is denoted by adj A 3. If A is a matrix of order n x n then i) adj ( KA) n 1 K adj A ii) adj A n 1 A iii) adj ( adj A) n A iv) Adj ( adj A) ( n 1) A 4. Inverse of a matrix: Let a be a non singular matrix if there exist a matrix B such that AB BA I then B is called the inverse of A denoted by A 1 5. Matrix inverse if exist is unique 6. If A is a non singular matrix then 1 1 AA A A I heorem 1: If A is non singular matrix then transpose of matrix. 1 1 ( A ) ( A ) where A is the Sol: 1 1 AA A A I 1 I 1 ( AA ) ( I) A ( A ) I (1) 1 I 1 ( A A) ( I) A ( A ) I () From (1) and () we have 1 1 A ( A ) ( A ) A I his is in the form AB BA I B A 1 Hence ( A ) ( A ) 1 1

9 heorem : If A, B are two non singular matrices then prove that ( AB) B A Sol: A is a non singular matrix 1 1 AA A A I (1) B is a non singular matrix 1 1 BB BB I () Let AB P; B A 1 1 Q PQ ( AB)( B A ) A( BB ) A A( IA ) AA I QP ( B A )( AB) A( BB ) A B ( IB) B B I PQ QP I Q P 1 B A ( AB) heorem 3: If A is a non singular matrix then A 1 adj A det A Sol: Let a1 b1 c1 A a b c a 3 b3 c 3 A1 A A3 adj A B B B 1 3 C 1 C C 3 Where A1, B1, C 1 are the cofactors of a1, b1, c 1 A, B, C are the cofactors of a, b, c A3, B3, C 3 are the cofactors of a3, b3, c 3

10 a b c A A A A( adj A) a b c B B B a 3 b3 c 3 C1 C C 3 a1a1+ bb 1 1+ cc 1 1 a1a + bb 1 + cc 1 a1a3 + bb cc 1 3 aa1 bb1 cc1 aa bb cc aa3 bb3 cc a 3A1+ b3b1+ c3c1 a3a + b3b + c3c a3a3 + b3b3 + c3c 3 def A def A 0 0 def A adj A A( adj A) (det A) I A I def A Similarly we can prove adj A A I def A adj A adj A A A I det A det A adj A A det A 1 Sub Matrix: A matrix obtained by eliminating some row or columns (or both) of a matrix is called sub matrix r rowed minor : he determinant of square sub matrix of order r is called r-rowed minor Rank of a matrix: A positive integer r is said to be the rank of the matrix of there exist i) At least one non-zero r-rowed minor ii) Every (r + 1) rowed minor is zero iii) rank of null matrix is O

11 Elementary transformation : 1. Interchanging any two row (or column). Multiplication of elements of a row (or column) 3. Subtracting from (adding to) the elements of one row, the corresponding elements of any other row multiplied by a non zero number. Echelon form: A matrix A is said to be in Echelon form if the number of zeros before the first non zero element in a row is less than the number of such zeros in the net row Ex: A Equivalent matrices : wo matrices A and B are called equivalent if one can be btained from the other by a finite number of elementary transformation it is denoted by A B he equations ax 1 + by 1 + ax d1; ax + by + cz d And a3x+ b3y+ c3z d3 are called system of linear equations he equations can be expressed in the matrix form as Ax B a1 b1 c1 x d1 A a b c x y B d a 3 b3 c 3 z d 3 A is called coefficient matrix X is called variable matrix B is called constant matrix he matrix a1 b1 c1 d1 a b c d a 3 b3 c3 d 3

12 Is called augmented matrix denoted by k If rank of A rank of k no. of variables hen system of equations has unique solutions Rank of A rank of k no. of variables the system of equations have infinite solutions If rank of A rank of k then the system has no solution. A system of equations is said to be consistent if it has a solution A system of equations is said to be inconsistent if it has no solution Matrix inversion method of solving the equation he matrix form of equations is Ax B If A 0 then x 1 A B Cramer s rule a b c a b c a b c d d c d d c 1 d d c a d c a d c a d c a b d a b d 3 a b d 3 3 3

13 x 1 ; y 3 z Gauss Jordan method Augmented matrix a1 b1 c1 d1 a b c d a 3 b3 c3 d 3 By applying finite no. of row transformations the matrix will be transformed into α β γ x α, y β, z γ

14 EXERCISE 3(A) I. 1. Write the following as a single matrix. Sol. i) [ 1 3] + [0 0 0] [ ] [ 1 3] ii) iii) iv) If A , B 3 5 and X values of x 1, x, x 3 and x 4. Sol. A + B X x1 x x3 x x x x3 x 4 x 1,x 4,x 7,x x x x 1 x 3 4 and A + B X, then find the

15 If A 1 4, B 0 and C Sol. A + B + C then find A + B + C If A 0, B 1 3 and X A + B then find the matrix X Sol. X A + B X If Sol. x3 y8 5 z+ 6 a4, find the values of x, y, z and a. x3 y8 5 Given z+ 6 a4 x 3 5 x y 8 y y 5 z + z 4 a 4 6 a

16 II. 1. x1 5y 1 3 If 0 z a then find the values of x, y, z and a. Sol. x1 5y 1 3 Given 0 z a x 1 1 x y 3 y 5 3 z 1 4 z a 5 0 a Find trace of A if A Sol. race of A Sum of the diagonal elements If A 3 4 and B find B A and 4A 5B Sol. Given A 3 4, B B A A 5B

17 If A 3 1 and B 1 3 find 3B A Sol. A 3 1, B B A EXERCISE-3(B) I. 1. Find the following products wherever possible. Hint: (1 3) by (3 1) Sol. i) [ 1 4 ] 1 [ ] [ ] [ ] ii) ( )

18 iii) iv) v) First matrix is a 3 3 matrix and second matrix is 3 matrix. Number of columns in first matrix Number of rows in second matrix. Matrix product is not possible vi) Number of columns in first matrix 1 Number of rows in second matrix Number of columns in first matrix Number of rows in second matrix Multiplication of matrices is not possible vii)

19 0 c b a ab ac viii) c 0 a ab b bc b a 0 ac bc c 0+ abcabc b cb c bc bc a c + a c abc + abc ac + ac a b a b ab ab abc abc If A 4 5 and B 4 5, do AB and BA exist? If they exist, find 1 them. Do A and B commutative with respect to multiplication of matrices Sol. Given A 4 5 and B AB BA AB BA A and B are not commutative with respect to multiplication of matrices. 3. Find A 4 where A 1 1 Sol. A 4 4 A.A

20 i 0 4. If A 0 i, find A. Sol. A i 0 i 0 i 0 A, A 0 i 0 i 0 i i i 5. If A 0 i, B 1 0 and C i 0 then show that (i) A B C I, (ii) AB BA C (i 1 and I is the unit matrix of order ) Sol. i) A i 0 i 0 A.A 0 i 0 i i i B B.B I C C.C 0 i 0 i i 0 i 0 i I 0 i A B C 1 i ii) AB 0 i i 0 i C i 0 i i 0 0 i BA C i i 0 AB BA C.

21 If A 1 3 and B 1 0 4, find AB. Find BA if exists Sol. Given A 1 3, B AB Order of AB is 3 BA does not exist since number of columns in B No.of rows in A If A 1 k and A 0, then find the value of k. Sol. A k 1 k k 0 0 k 4 k k 0 4k 8 k II If A then find A Note : A is diagonal matrix. a 0 0 Sol. If A 0 b 0, then 0 0 c n a 0 0 n n A 0 b 0,n N n 0 0 c A

22 If A 5 6 then find A Sol. A A.A A A A If A 0 1 1, then find A 3 3A A 3I Sol Given A A A.A

23 A 3 A A Now A 3 3A A 3I O A 3 3A A 3I If I 0 1 and E 0 0, show that (ai + be) 3 a 3 I + 3a be a b Sol. ai + be a b a (ai + be) a b a b a ab 0 a 0 a 0 a 3 (ai + be) 3 a ab a b a 3a b 0 a 3 0 a 0 a

24 3 a 0 0 3a b a a 3a b ai+ 3abE III. 1. If A a a 0, then for any integer n 1 show that A n 0 0 a3 a1 0 0 Sol. Given A 0 a a3 We shall prove the result by Mathematical induction. n a1 0 0 A n n 0 a 0 n 0 0 a3 When n 1 a1 0 0 A 1 0 a a3 he result is true for n 1. Suppose the result is true for n k k a1 0 0 k k i.e. A 0 a 0 k 0 0 a3 k 1 k Now A + A A k a1 0 0 a1 0 0 k 0 a 0 0 a 0 k 0 0 a 0 0 a 3 3 n a1 0 0 n 0 a 0. n 0 0 a3

25 k a1 a k a a k a3 a3 k+ 1 a1 0 0 k+ 1 0 a 0 k a3 he given result is true for n k + 1 By Mathematical induction, the given result is true for all positive integral values of n. n a1 0 0 n n i.e. A 0 a 0, for any integer n 1. n 0 0 a3. If θ φ π, show that cos θ cosθsin θ cos φ cos φsin φ 0 cos θsin θ sin θ cosφsin φ sin φ π π Sol. Given θφ θ +φ π cos θ cos +φ sinφ π sin θ sin +φ cos φ cos θ cos θsin θ cos θsin θ sin θ sin φ sin φcos φ sin φcos φ cos φ cos θ cos θsin θ cos θsin θ sin θ cos φ cos φsin φ cos φsin φ sin φ

26 sin φ sin φcos φ cos φ cos φsin φ sin φcosφ cos φ cosφsin φ sin φ 3 3 sin φcos φ sin φcos φ sin φcos φ sin φcos φ 3 3 sin φcos φ+ sin φcos φ sin φcos φ+ sin φcos φ If A 1 1 then show that A n 1+ n 4n n 1 n, n is a positive integer. Sol. We shall prove the result by Mathematical Induction. n 1+ n 4n A n 1 n n 1 A he result is true for n 1 Suppose the result is true for n k k 1+ k 4k A k 1 k k+ 1 k 1+ k 4k 3 4 A A A k 1 k k4k 4 8k+ 4k 3k 1 k 4k 1 k + + k + 3 4k 4 k 1 k (k+ 1) 4(k+ 1) k+ 1 1 (k+ 1) he given result is true for n k + 1 By Mathematical Induction, given result is true for all positive integral values of n. 4. Give examples of two square matrices A and B of the same order for which AB 0 and BA 0. a Sol. A a 0, B a a a hen AB 0 a 0 a a a BA a a a 0 a + a 0+ 0

27 0 0 0 a 0 AB 0 and BA 0 EXERCISE 3(C) If A and B 0 1 then find (AB ). 1 0 Sol. B AB (AB ) If A and B 4 0 find A + B and 3B A Sol. A 5 0 A B B

28 4 4 A + B B B A If A 5 3 then find A + A and A.A. 4 Sol. A A A+ A

29 If A 5 6 is a symmetric matrix, then find x. 3 x 7 Sol. A is a symmetric matrix A A x 7 3 x 7 Equating nd row, 3 rd column elements we get x If A 0 is a skew symmetric matrix, find x. 1 x 0 Hint : A is a skew symmetric matrix A A Sol. A is a skew symmetric matrix A A x x 0 1 x 0 Equating second row third column elements we get x Is symmetric or skew symmetric Sol. Let A A A A is a skew symmetric matrix.

30 II. cos α sin α 1. If A sin α cos α, show that A A A A I. cos α sin α cos α sin α Sol. A A sin α cos α sin α cos α cos α+ sin α sin αcos α+ sin αcos α sin αcosα+ cos αsin α sin α+ cos α 1 0 I...(1) 0 1 cos α sin α cos α sin α A A sin α cos α sin α cos α cos α+ sin α cos αsin αsinαcos α sin αcosαcos αsin α sin α+ cos α 1 0 I...() 0 1 From (1), () we get A A A A I.. If Sol A 4 0 and B B A 4B B 0 5 then find 3A 4B. 1 0

31 A 1 and Sol. B B 4 then find AB and BA B AB A A

32 BA For any square matrix A, show that AA is symmetric. Sol. A is a square matrix (AA ) (A ) A A A (AA ) AA AA is a symmetric matrix.