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4., MATRICES, MATRICES (A Review), • Definitions, 1. Matrix : A set of mn numbers of a field F (real or complex), arranged in the form, of a rectangular array having m rows and n columns in called an m xn matrix, over the field F., 2. Square Matrix: A , matrix is said to be a square matrix if m = n. In other, words, a matrix is a square matrix if the number of rows number of columns., 3. Rectangular Matrix : A, matrix is said to be rectangular matrix if m # n., m x n, 4. Diagonal Matrix : A square matrix A = [a] is said to be a diagonal matrix if all, its non-diagonal elements are zero. The diagonal entries may or may not be zero., %3D, 5. Unit (or Identity) Matrix : A diagonal matrix is said to be a unit matrix or an, identity matrix if each of its diagonal element is equal to unity (one)., 6. Null Matrix : The m x n matrix whose elements are all zero is called the null, matrix or zero matrix., • Transpose of a Matrix, Let A = lam x n', then the matrix of the typen x m obtained by interchanging its rows, %3D, and columns is called the transpose of A and is denoted by A' or AT.ad, Then A' = [al, x m where a, = a, i.e., the (i, j)th element of A' = (j, i)th element of A, for all values of i and j., %3D, %3D, %3D, Theorem : If A', B' denote the transposes of A, B respectively, then, (a) (A')' = A, (b) (A+ B) A' +B', (c) (kA) = kA'; k being a scalar, (d) (AB)' = B'A', %3D, • Conjugate of a Matrix, Let A = [am xn over the complex number system, then conjugate of A i.e., A can be, obtained by replacing each of its elements by their corresponding complex conjugate., In general if A (a,), then A =, la,l, where, = a; i.e., (i, j)th element of, complex conjugate of the (i, j)th element of A., %3D

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4.2, RLNS, MATRICES, 4.3, (6) (A + B) =A+ B, 5. If A is a non-singular matrix, then det. (A-) (det. A-., 6. The adjoint of a non-singular matrix is non-singular., 7. If A is non-singular matrix of order n, then, (d) (AB) = A.B, (a) (A) = A, (c) (kA) = A, when k is any scalar, (a) adj. A = 1A|-1, (b) adj.(adj. A) = |A-2, A, (c) adj. (AB) = adj. B. adj. A, if A and B are square matrices of same order., • Transposed Conjugate of a Matrix, (d) Jadj. (adj. A) |adj. A/a-1= JA|-17, In general, adj adj .. adj A = (|A|-1, r times, (b) (A + B)® = A° + Be, (e) (adj. Aym = adj. Am, (a) (A )® = A, O adj. O = 0, where O is a null matrix., (c) (kA) = EA": when & is any complex number, g) adj I, = L, (h) Adjoint of symmetric matrix is also symmetric matrix., (d) (AB = B. A°., • Adjoint of a Square Matrix, (i) Any non-singular symmetric matrix has symmetric inverse., ) If B is non-singular square matrix then |A| = |B AB|., Let A = la, Ais called the adjoint of A and is denoted by adj. A., 1o 8. Transpose of a non-singular matrix is non-singular and (A and (A-Y., Results :, SYMMETRIC MATRIX, (a) IFA be a square matrix of the type n x n, then A (adj. A) = (adj. A) . A = JAu, where I is a unit matrix of order n., A square matrix A = [a,] is said to be symmetric if A' = A or a =a, for all i and j., (6) If A and Bare two matrices of same order n, then adj. (AB) = (adj. B) (adj. A)., SKEW-SYMMETRIC MATRIX, • Inverse of a Square Matrix, A square matrix A = [a] is said to be skew-symmetric if A' =-A, If A is an i rowed square matrix and if there exists another n-rowed square matri, B such that AB = BA = I, then A is said to be invertible and B is called the inverse d, or a, =-a, for all i and j., A., RESULTS OF SYMMETRIC AND SKEW-SYMMETRIC MATRICES, • Singular and Non-singular Matrices, If A and B are symmetric matrices then AB is symmetric iff AB = BA., A square matrix A is said to be singular if JAJ = 0 and non-singular if |A| + 0., If A is a square matrix, then, • Theorems :, (a) A + A' is symmetric, (b) A- A' is skew-symmetric, Every square matrix can be expressed in one and only one way as the sum of a, symmetric and skew-symmetric matrices., det. A 0. i.e., A is non-singular., a, 2. If A is invertible then Al is unique and A =, The diagonal elements of a skew-symmetric matrix are all zero., Every skew-symmetric matrix of odd order is a singular matrix., adj. A, If A is a symmetric or a skew-symmetric matrix, then kA is also symmetric or, skew-symmetric matrix respectively., (AB)" = B-1 A-1,, 4. IfA is invertible, so is A and (A--=A., The sum of two symmetric or skew-symmetric matrix are also symmetric or, skew-symmetric matrix respectively., п,, conjugate of A and is by A. In other words, A° = (A, Theorem : If of A, B then, Theorem: IfA, B® the of A, B then, Let A = la then the of the matrix A is transpo

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TRAN, 4.4, 4.5, MATRICES, The sum of two Hermitian or Skew-Hermitian matrices is also Hermitian or, or skew, Skew-Hermitian respectively., symmetric., symmetric, If A and B are Hermitian, then AB is Hermitian iff AB = BA., Imaginary part in principal diagonal element of a Hermition matrix is zero., Determinant of Hermitian matrix is always real value., a, Determinant of Skew-Hermitian matrix of odd order is either zero or purely imaginary, and even order is always, real., skew-symmetric according as n is odd or even., Inverse of Hermitian matrix is Hermitian., Inverse of Skew-Hermitian matrix is Skew-Hermitian., and A" is symmetric if n is positive even integer., a perfer, B0INTAM YAATIWU GMA, ORTHOGONAL AND UNITARY MATRICES, square., skew-symmetric., ORTHOGONAL MATRIX, A square matrix A over the field of reals is said to be orthogonal if AA' = A'A = I, A' = A-1., HERMITIAN MATRIX, A square matrix A = [a over complex numbers is said to be Hermiti, a = a, for all i and j or in other words A" = A i.e., transposed conjugate of the matriri, %3D, or, An orthogonal matrix A is said to be proper if A| = 1 and improper if |A = -1., equal to the matrix itself., PROPERTIES OF ORTHOGONAL MATRIX, I SKEW-HERMITIAN MATRIX, A square matrix A [a] over complex numbers is said to be Skew-Hermitian if, 1. The determinant of an orthogonal matrix is + 1 or every orthogonal matrix is non-, singular., a, = - a for all i and j or in other words A' = - A i.e., transposed conjugate of the, 2. The product of two orthogonal matrices of the same order is orthogonal., matrix is equal to the negative of the matrix itself., 3. The inverse and transpose of an orthogonal matrix are orthogonal., RESULTS, The diagonal elements of a Hermitian matrix are all real., UNITARY MATRIX, The diagonal elements of a Skew-Hermitian matrix are either zero or puren, imaginary., A square matrix A over the field of complex numbers is said to be unitary if, A.A" = A°.A =I i.e., A° = A-1, If A is a square matrix, then A+ A is Hermitian and A - A9 is Skew-Hermitian., PROPERTIES OF UNITARY MATRICES, Skew-Hermitian matrix., 1. A real matrix is unitary iff it is orthogonal., If A is Hermitian then, iA is Skew-Hermitian., 3. Product of two unitary (orthogonal) matrices is unitary (orthogonal) matrices., 4. If A is unitary, then |A = ± 1, 2. The transpose and inverse of a unitary matrix is also unitary., Hermitian matrices., Be AB is Hermitian or Skew-Hermitian according, 5. If A is Hermitian and P is unitary, then P-AP is Hermitian,, Skew-Hermitian., 6. If A is unitary and Hermitian matrix, then A is involutory (i.e., A I), as A is Hermitian of, • If A is any square matrix, then AA' and A'A are both symmetric., B'AB is or skew-symmetric as A is c, If A is a adj. A is also symmetric., • If A is then A" is where n is any positive integer,, • The of a matrix of order is always:, If A and B are then AB + BA is and AB ake, If A is a matrix of n, then adj. A is o

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4.6, DIAGONAL MATRIX, MATRICES, 1. Number of elements along principal diagonal is equal to n., 4.7, ELEMENTARY MATRIX, 2. Number of elements along super diagonal line is n- 1., A matrix obtained from an identity matrix I, by a single elementary transformation is, called an elementary matrix or E-matrix., 3. Number of elements along sub diagonal line is n- 1., TRIANGULAR MATRIX, NORMAL MATRIX, A matrix A is said to be normal if AA° = A A., n (n +1), Every unitary, real orthogonal, Hermitian, Skew-Hermitian, real Skew-Hermitian matrices, 2., are normal., 2. Number of atleast zeros of upper triangular matrix of ordern is n (n - 11, 2, SELF CONJUGATE MATRIX, A matrix A is said to be self conjugate matrix if A = A°., 3. Number of non-trivial elements in symmetric matrix of order n is n (n4 1, 2, INVOLUTORY MATRIX, A square matrix A is said to be involutory if A = I., Property :, 1. Even power of involutory matrix is identity matrix., 2, 2. Every odd power of involutory matrix is matrix itself., IDEMPOTENT MATRIX, 3. Every involutory matrix is expressible as a difference of idempotent matrices, 1. A square matrix A is an idempotent matrix if A? = A., 2. Any idempotent matrix has its determinant values 0 or 1., i.e.,, A =, 1, (A + I) –, 1, (I- A) = A* - A, 3. Every square matrix having determinant values 0 or 1 need not be an jidempt, matrix., RANK OF A MATRIX, 4. If A is idempotent then A" = A where k is positive integral value greater than 1., ELEMENTARY ROW OPERATIONS ON A MATRIX, 5. If AB = A and BA = B, then A and B are idempotent., 6. If A and B are idempotent matrices, then (A + B) is also idempotent if A and B are, anti-commute i.e., (A + B)? = (A + B),, Let A be an m xn matrix over a field F. The three elementary row operations on A are, defined as:, 1. Multiplication of any row of A by a non-zero element c of F., 7. Product of commutative idempotent matrices is idempotent., 8. Every identity and null matrices are idempotent., 2. Addition to the elements of any row of A the corresponding elements of any other, row of A multiplied by any element a of F., I NILPOTENT MATRIX, 3. Interchanging of two rows of A., SUBMATRICES, A non-zero square matrix is said to be nilpotent if there exist a positive integer* o, that A = 0., 1. Maximum number of submatrices of a matrix of order mn in which all the elements, are different = (2m - 1) (2" - 1)., The least positive value of k for which Ak = 0 is called index of the matrix., 2. Minimum number of submatrices of a matrix of order m x n in which all the elements, are equal = m x n., • Determinant of a nilpotent matrix is zero., 3. Maximum number of proper submatrices of a matrix of order m x n (all element are, different) are (2m – 1) (2" – 1)- 1., TRI DIAGONAL MATRIX, 4. Minimum number of submatrices of a matrix of order m x n if all the elements are, same = mn - 1., 4. Number of in matrix of n is n (n- 1)

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4.8, MINORS OF A MATRIX, MATRICES, 2. All the non-zero rows, precede the zero rows, if any., RANK OF A MATRIX, 3. The number of zeros before the leading entry in each row is less than the number of, such zero's in the succeeding rows., A non-zero matrix A, of the type m xn is said to have rankr if, 1. It has atleast one non-zero minor of order r., 1521, 2 Every minor of order r+l of A, if any is zero., The rank of a matrix A is denoted by p A)., 1 25, 017 and, 0 179, For exampe,, are in row-echelon form, 0018, 001, 0000, RESULTS, ROW RANK AND COLUMN RANK OF A MATRIX, The number of non-zero rows in the row echelon form of a matrix A is called the, row rank of A and is denoted by pa A)., diagonal, • Ifl is an identity matrix of order n, then p (I) = n, • The number of non-zero rows in the column echelon form of a matrix A is called the, column rank of A and is denoted by p(A, Ifp (A) = n, then p (AB) = n, if B is non-singular., p(A)=p(A'), • PLAB)Sp(A) and p(AB)Sp (B), • p (AB) = p(BA), The number of non-zero rows in a triangular form also gives the rank of the given, matrix., NORMAL FORM (CANONICAL FORM) OF A MATRIX, • No skew-symmetric matrix can be of rank 1., .IfA is a non-zero column matrix and B is a non-zero matrix, then p (AB) = 1., Every non-zero matrix of order m x n can be reduced by means of elementary row and, column operations into equivalent matrix of any of the following forms:, IfA is any n square matrix of rank n-1, then adj. A + 0, Rank of a matrix does not change by applying any elementary operations., All elementary matrices are non-singular., I,, : 0, (i), () IL : 01, (i) [11, (i), Rank of a matrix unchanged when it is multiplied by a non-singular matrix., Rank of a skew symmetric matrix can not be equal to 1., p (A) + pI-A) =n, where n is order of A., Ifp (A) = n, thenp (adj. A) = n-1, where I is the identity matrix of orderr and O represents zero matrix of any order which, is called its normal form or canonical form., Remark : The rank of an mxn matrix A is r if and only if it can be reduced to the, 1, 0, form, by a finite chain of E-operations., If p (A) = n-1, then p ladj. A)= 1, If p (A) = n-2, n-3, , then p (adj. A) 0, p(A) + p (B)-n Sp(AB)s min (p(A), p(B), where n is the order of A and D., LINEAR DEPENDENCE AND INDEPENDENCE, The vectors x,, X, , x are said to be linearly dependent, if there exists a scalars, C, not all zero such that c, + Ct, + .. tex = 0 and if all the scalars, respectively., C, Co, ,, ROW ECHELON MATRIX, C1, Cas C, are zero, then x, , , x are called linearly independent., If two vectors are linearly dependent, then one of them is the scalar multiple of the, other., RESULTS, conditions are satisfied:, Every super set of a linearly dependent set is linearly dependent., Every subset of a linearly independent set of vector is linearly independent., entry of the row.