MATRICES-(E)-PART-I-04-THEORY

8.2.1 Definition . A rectangular arrangement of numbers (which may be real or complex numbers) in rows and columns, is called a matrix. This arrangement is enclosed by small ( ) or big [ ] brackets. The numbers are called the elements of the matrix or entries in the matrix. A matrix is represented by capital letters A, B, C etc. and its elements by small letters a,b,c,x,y etc. The following are some examples of matrices: , , , 8.2.2 Order of a Matrix. A matrix having m rows and n columns is called a matrix of order m×n or simply m×n matrix (read as 'an m by n matrix). A matrix A of order m×n is usually written in the following manner , where Here denotes the element of ith row and jth column. Example : order of matrix is 2×3 Note :  A matrix of order m×n contains mn elements. Every row of such a matrix contains n elements and every column contains m elements. 8.2.3 Equality of Matrices . Two matrix A and B are said to be equal matrix if they are of same order and their corresponding elements are equal Example: If and are equal matrices. Then 8.2.4 Types of Matrices. (1) Row matrix : A matrix is said to be a row matrix or row vector if it has only one row and any number of columns. Example : [5 0 3] is a row matrix of order 1× 3 and [2] is a row matrix of order 1×1. (2) Column matrix : A matrix is said to be a column matrix or column vector if it has only one column and any number of rows. Example : is a column matrix of order 3×1 and [2] is a column matrix of order 1×1. Observe that [2] is both a row matrix as well as a column matrix. (3) Singleton matrix : If in a matrix there is only one element then it is called singleton matrix. Thus, is a singleton matrix if Example : [2], [3], [a], [–3] are singleton matrices. (4) Null or zero matrix : If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by O. Thus is a zero matrix if for all i and j. Example : are all zero matrices, but of different orders. (5) Square matrix : If number of rows and number of columns in a matrix are equal, then it is called a square matrix. Thus is a square matrix if . Example : is a square matrix of order 3×3 (i) If then matrix is called a rectangular matrix. (ii) The elements of a square matrix A for which are called diagonal elements and the line joining these elements is called the principal diagonal or leading diagonal of matrix A. (iii) Trace of a matrix : The sum of diagonal elements of a square matrix. A is called the trace of matrix A , which is denoted by tr A. Properties of trace of a matrix : Let and and be a scalar (i) (ii) (iii) (iv) or (v) (vi) tr (0)= 0 (vii) (6) Diagonal matrix : If all elements except the principal diagonal in a square matrix are zero, it is called a diagonal matrix. Thus a square matrix is a diagonal matrix if when . Example : is a diagonal matrix of order 3×3, which can be denoted by diag [2, 3, 4] Note :  No element of principal diagonal in a diagonal matrix is zero.  Number of zeros in a diagonal matrix is given by where n is the order of the matrix.  A diagonal matrix of order having as diagonal elements is denoted by . (7) Identity matrix : A square matrix in which elements in the main diagonal are all '1' and rest are all zero is called an identity matrix or unit matrix. Thus, the square matrix is an identity matrix, if We denote the identity matrix of order n by . Example : [1], are identity matrices of order 1, 2 and 3 respectively. (8) Scalar matrix : A square matrix whose all non diagonal elements are zero and diagonal elements are equal is called a scalar matrix. Thus, if is a square matrix and , then A is a scalar matrix. Example : [2], are scalar matrices of order 1, 2 and 3 respectively. Note :  Unit matrix and null square matrices are also scalar matrices. (9) Triangular Matrix : A square matrix is said to be triangular matrix if each element above or below the principal diagonal is zero. It is of two types (i) Upper Triangular matrix : A square matrix is called the upper triangular matrix, if when . Example : is an upper triangular matrix of order 3×3. (ii) Lower Triangular matrix : A square matrix is called the lower triangular matrix, if when i< j. Example : is a lower triangular matrix of order 3×3. Note :  Minimum number of zeros in a triangular matrix is given by where n is order of matrix.  Diagonal matrix is both upper and lower triangular.  A triangular matrix is called strictly triangular if for Example: 1 A square matrix in which for and (constant) for is called a [IIT Screening 1990] (a) Unit matrix (b) Scalar matrix (c) Null matrix (d) Diagonal matrix Solution: (b) When for and is constant for then the matrix is called a scalar matrix Example: 2 If A, B are square matrix of order 3, A is non singular and then B is a (a) Null matrix (b) Singular matrix (c) Unit matrix (d) Non singular matrix Solution: (a) AB = 0 when B is null matrix. Example: 3 The matrix is known as (a) Symmetric matrix (b) Diagonal matrix (c) Upper triangular matrix (d) Skew symmetric matrix Solution: (c) We know that if all the elements below the diagonal in a matrix are zero, then it is an upper triangular matrix. Example: 4 In an upper triangular matrix n×n, minimum number of zeros is (a) (b) (c) (d) None of th Solution: (a) As we know a square matrix is called an upper triangular matrix if for all i>j . Number of zeros = Example: 5 If is a scalar matrix then trace of A is (a) (b) (c) (d) Solution: (d) The trace of Sum of diagonal elements. 8.2.5 Addition and Subtraction of Matrices. If and are two matrices of the same order then their sum A+B is a matrix whose each element is the sum of corresponding elements. i.e. Example : If and , then Similarly, their subtraction is defined as i.e. in above example Note :  Matrix addition and subtraction can be possible only when matrices are of the same order. Properties of matrix addition : If A, B and C are matrices of same order, then (i) (Commutative law) (ii) (Associative law) (iii) where O is zero matrix which is additive identity of the matrix. (iv) , where is obtained by changing the sign of every element of A, which is additive inverse of the matrix. (v) (Cancellation law) 8.2.6 Scalar Multiplication of Matrices. Let be a matrix and k be a number, then the matrix which is obtained by multiplying every element of A by k is called scalar multiplication of A by k and it is denoted by kA. Thus, if , then . Example : If , then Properties of scalar multiplication: If B are matrices of the same order and are any two scalars then (i) (ii) (iii) (iv) Note :  All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars. 8.2.7 Multiplication of Matrices. Two matrices A and B are conformable for the product AB if the number of columns in A (pre-multiplier) is same as the number of rows in B (post multiplier).Thus, if and are two matrices of order m×n and respectively, then their product AB is of order and is defined as = (ith row of A)(jth column of B) .....(i), where i=1, 2, ..., m and j=1, 2, ...p Now we define the product of a row matrix and a column matrix. Let be a row matrix and be a column matrix. Then …(ii). Thus, from (i), Sum of the product of elements of ith row of A with the corresponding elements of jth column of B. Properties of matrix multiplication If A,B and C are three matrices such that their product is defined, then (i) (Generally not commutative) (ii) (Associative Law) (iii) , where I is identity matrix for matrix multiplication (iv) (Distributive law) (v) If (Cancellation law is not applicable) (vi) If AB= 0 It does not mean that A= 0 or B = 0, again product of two non zero matrix may be a zero matrix. Note :  If A and B are two matrices such that AB exists, then BA may or may not exist.  The multiplication of two triangular matrices is a triangular matrix.  The multiplication of two diagonal matrices is also a diagonal matrix and diag  The multiplication of two scalar matrices is also a scalar matrix.  If A and B are two matrices of the same order, then (i) (ii) (iii) (iv) (v) 8.2.8 Positive Integral Powers of A Matrix. The positive integral powers of a matrix A are defined only when A is a square matrix. Also then , . Also for any positive integers m ,n. (i) (ii) (iii) (iv) where A is a square matrix of order n. 8.2.9 Matrix Polynomial . Let be a polynomial and let A be a square matrix of order n. Then is called a matrix polynomial. Example : If is a polynomial and A is a square matrix, then is a matrix polynomial. Example: 6 If (a) (b) (c) (d) Solution: (c) Since Example: 7 If and then (a) (b) (c) (d) Solution: (b) . On comparing, we get, Example: 8 If , then equals (a) (b) (c) (d) Solution: (a) , ; Example: 9 If and then value of a and b are (a) (b) (c) (d) Solution: (b) We have ∵  . On comparing, we get,  and  Example: 10 The order of is (a) 3×1 (b) 1×1 (c) 1×3 (d) 3×3 Solution: (b) Order will be Example: 11 Let . Then is equal to [AMU 1995] (a) (b) (c) (d) Solution: (c) We have , Example: 12 For the matrix , which of the following is correct [Kerala (Engg.) 2001] (a) (b) (c) (d) Solution: (b) ,  Example: 13 If , then the value of for which is (a) 1 (b) ¬–1 (c) 4 (d) No real values Solution: (d) ∵ (given) Then  and . Clearly no real value of 8.2.10 Transpose of a Matrix. The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called transpose of Matrix A and is denoted by or . From the definition it is obvious that if order of A is m×n, then order of is n×m Example : Transpose of matrix is Properties of transpose : Let A and B be two matrices then (i) (ii) and B being of the same order (iii) be any scalar (real or complex) (iv) and being conformable for the product AB (v) (vi) 8.2.11 Determinant of a Matrix . If be a square matrix, then its determinant, denoted by |A| or Det (A) is defined as Properties of determinant of a matrix (i) exists A is square matrix (ii) (iii) (iv) if A is a square matrix of order n (v) If A and B are square matrices of same order then |AB|=|BA| (vi) If A is a skew symmetric matrix of odd order then (vii) If then (viii) Example: 14 If A and B are square matrices of same order then [Pb. CET 1992; Roorkee 1995; MP PET 1990; Rajasthan PET 1992, 94] (a) (b) (c) (d) Solution: (b) , , where = Alternatively, Let ; …..(i) and …..(ii) From (i) and (ii), Example: 15 If A,B are 3×2 order matrices and C is a 2×3 order matrix, then which of the following matrices not defined [ (a) (b) (c) (d) Solution: (a) Order of A is 3 × 2 and order of B is 3 × 2 and order of is 2 × 3 then + is not possible because order are not same. 8.2.12 Special Types of Matrices. (1) Symmetric and skew-symmetric matrix (i) Symmetric matrix : A square matrix is called symmetric matrix if for all i, j or Example : Note :  Every unit matrix and square zero matrix are symmetric matrices.  Maximum number of different elements in a symmetric matrix is (ii) Skew-symmetric matrix : A square matrix is called skew- symmetric matrix if for all i, j or . Example : Note :  All principal diagonal elements of a skew- symmetric matrix are always zero because for any diagonal element.  Trace of a skew symmetric matrix is always 0. Properties of symmetric and skew-symmetric matrices: (i) If A is a square matrix, then are symmetric matrices, while is skew- symmetric matrix. (ii) If A is a symmetric matrix, then are also symmetric matrices, where , and B is a square matrix of order that of A (iii) If A is a skew-symmetric matrix, then (a) is a symmetric matrix for , (b) is a skew-symmetric matrix for , (c) kA is also skew-symmetric matrix, where , (d) is also skew- symmetric matrix where B is a square matrix of order that of A. (iv) If A, B are two symmetric matrices, then (a) are also symmetric matrices, (b) is a skew- symmetric matrix, (c) AB is a symmetric matrix, when . (v) If A,B are two skew-symmetric matrices, then (a) are skew-symmetric matrices, (b) is a symmetric matrix. (vi) If A a skew-symmetric matrix and C is a column matrix, then AC is a zero matrix. (vii) Every square matrix A can uniquelly be expressed as sum of a symmetric and skew-symmetric matrix i.e. . (2) Singular and Non-singular matrix : Any square matrix A is said to be non-singular if and a square matrix A is said to be singular if . Here (or det(A) or simply det |A| means corresponding determinant of square matrix A. Example : then is a non singular matrix. (3) Hermitian and skew-Hermitian matrix : A square matrix is said to be hermitian matrix if . Example : are Hermitian matrices. Note :  If A is a Hermitian matrix then is real thus every diagonal element of a Hermitian matrix must be real.  A Hermitian matrix over the set of real numbers is actually a real symmetric matrix and a square matrix, A=|aij| is said to be a skew-Hermitian if . Example : are skew-Hermitian matrices.  If A is a skew-Hermitian matrix, then i.e. must be purely imaginary or zero.  A skew-Hermitian matrix over the set of real numbers is actually a real skew-symmetric matrix. (4) Orthogonal matrix : A square matrix A is called orthogonal if i.e. if Example : is orthogonal because In fact every unit matrix is orthogonal. (5) Idempotent matrix : A square matrix A is called an idempotent matrix if . Example : is an idempotent matrix, because . Also, are idempotent matrices because and . In fact every unit matrix is indempotent. (6) Involutory matrix : A square matrix A is called an involutory matrix if or Example : is an involutory matrix because In fact every unit matrix is involutory. (7) Nilpotent matrix : A square matrix A is called a nilpotent matrix if there exists a such that Example : is a nilpotent matrix because (Here P = 2) (8) Unitary matrix : A square matrix is said to be unitary, if I since and therefore if A=I, we have Thus the determinant of unitary matrix is of unit modulus. For a matrix to be unitary it must be non-singular. Hence (9) Periodic matrix : A matrix A will be called a periodic matrix if where k is a positive integer. If, however k is the least positive integer for which then k is said to be the period of A. (10) Differentiation of a matrix : If then is a differentiation of matrix A. Example : If then (11) Submatrix : Let A be m×n matrix, then a matrix obtained by leaving some rows or columns or both, of A is called a sub matrix of A. Example : If and are sub matrices of matrix (12) Conjugate of a matrix : The matrix obtained from any given matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex numbers is called conjugate of A and is denoted by . Example : then Properties of conjugates : (i) (ii) (iii) being any number (iv) and B being conformable for multiplication. (13) Transpose conjugate of a matrix : The transpose of the conjugate of a matrix A is called transposed conjugate of A and is denoted by The conjugate of the transpose of A is the same as the transpose of the conjugate of A i.e. . If then where i.e. the element of the conjugate of element of A. Example : If , then Properties of transpose conjugate (i) (ii) (iii) K being any number (iv) Example: 16 The matrix is known as (a) Upper triangular matrix (b) Skew-symmetric matrix (c) Symmetric matrix (d) Diagonal matrix Solution: (b) In a skew-symmetric matrix, and ,  each aii =0 Hence the given matrix is skew-symmetric matrix . Example: 17 The matrix is non singular if (a) (b) (c) (d) Solution: (a) The given matrix is non singular If |A|  0       Example: 18 The matrix is (a) Orthogonal (b) Involutary (c) Idempotent (d) Nilpotent Solution: (a) Since for given . For orthogonal matrix  . Similarly . Hence A is orthogonal Example: 19 If is symmetric, then x = (a) 3 (b) 5 (c) 2 (d) 4 Solution: (b) For symmetric matrix,    Example: 20 If A and B are square matrices of order n×n, then is equal to (a) (b) (c) (d) Solution: (d) Given A and B are square matrices of order n×n we know that Example: 21 If then which of the following statement is not correct (a) A is orthogonal matrix (b) is orthogonal matrix (c) Determinant (d) A is not invertible Solution: (d) therefore A is invertible. Thus (d) is not correct Example: 22 Matrix A is such that where I is the identity matrix. Then for (a) (b) (c) (d) Solution: (a) We have,  ;  Similarly and hence Example: 23 Let , the only correct statement about the matrix A is (a) (b) , where I is unit matrix (c) does not exist (d) A is zero matrix Solution: (a) . Also, exists as 8.2.13 Adjoint of a Square Matrix. Let be a square matrix of order n and let be cofactor of in A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A Thus, cofactor of in A. If then Where denotes the cofactor of in A. Example : Note :  The adjoint of a square matrix of order 2 can be easily obtained by interchanging the diagonal elements and changing signs of off diagonal elements.