Linear Algebra - Chapter 4 PDF

Chapter 4 Linear Algebra CHAPTER HIGHLIGHTS ☞ Introduction ☞ Systems of linear equations ☞ Determinants...

Chapter 4 Linear Algebra CHAPTER HIGHLIGHTS ☞ Introduction ☞ Systems of linear equations ☞ Determinants IntroDuction Column Matrix A matrix which has only one column A set of ‘mn’ elements arranged in the form of rectangular  a1  array having ‘m’ rows and ‘n’ columns is called an m × n   matrix (read as ‘m by n matrix’) and is denoted by A = [aij] a2 A =   or [aij ]n×1 where 1 ≤ i ≤ m; 1 ≤ j ≤ n      a11 a12 a13  a1n   an    a21 a22 a23  a2 n  or A= Diagonal Matrix A square matrix is said to be a diagonal         matrix if all its elements except those in the principal diago-  am1 am 2 am3  amn  nal are zeros. That is, if The element aij lies in the ith row and jth column. 1. m = n (A is a square matrix) and 2. aij = 0 if i ≠ j (The non-diagonal elements are zeros) Type of Matrices Square Matrix A matrix A = [aij]m×n is said to be a square A diagonal matrix of order ‘n’ with diagonal elements d1, matrix, if m = n (i.e., Number of rows of A = Number of d2,... , dn is denoted by Diag [d1 d2... dn]. columns of A) Scalar Matrix A diagonal matrix whose diagonal elements The elements a11, a22, a33,... , ann are called ‘DIAGONAL are all equal is called a scalar matrix. That is, if ELEMENTS’. The line containing the diagonal elements is the 1. m = n ‘PRINCIPAL DIAGONAL’. 2. aij = 0 if i ≠ j The sum of the diagonal elements of ‘A’ is the ‘TRACE’ 3. aij = k if i = j for some constant ‘k’. of A. Unit or Identity Matrix A scalar matrix of order ‘n’ in Row Matrix A matrix A = [aij]m×n is said to be row matrix, which each diagonal element is ‘1’ (unity) is called a unit if m = 1 (i.e., the matrix has only one row) matrix or identity matrix of order ‘n’ and is denoted by In. General form is A = [a1, a2,..., an] or [aij]1×n That is, Chapter 04.indd 71 5/19/2017 5:19:15 PM  2.72 | Part II Engineering Mathematics 1. m = n Properties of Transpose 2. aij = 0 if i ≠ j T − 1: (A′)′ = A, for any matrix A 3. aij = 1 if i = j T − 2: (A + B)′ = A′ + B′, for any two matrices A, B of 1 0 0 same order 1 0   T − 3: (KA)′ = KA′, for any matrix A Example: I1 = , I 2 =   , I3 =  0 1 0  0 1 0 0 1 T − 4: (AB)′ = B′A′, for any matrices A, B such that   number of columns of A = number of rows of B Null Matrix or Zero Matrix A matrix is a ‘null matrix’ or (REVERSAL LAW) zero matrix if all its elements are zeros. T − 5: (An)′ = (A′)n, for any square matrix A Upper Triangular Matrix A square matrix is said to be an Trace of a Matrix upper triangular matrix, if each element below the principal Let ‘A’ be a square matrix. The trace of A is defined as the diagonal is zero. That is, sum of elements of ‘A’ lying in the principal diagonal. 1. m = n Thus if A = [aij]n × n then trace of ‘A’ denoted by tr A = a11 2. aij = 0 if i > j + a22 +... + ann. 1 4 3 2 Properties of Trace of a Matrix Let A and B be any two   0 −1 6 1 square matrices and K any scalar then, For example,  0 0 3 2 1. tr(A + B) = trA + trB   0 0 0 9 4×4 2. tr(KA) = KtrA Lower Triangular Matrix A square matrix is said to be a 3. tr(AB) = tr(BA) lower triangular matrix, if each element above the principal diagonal is zero, i.e., if Conjugate of a Matrix A matrix obtained by replacing each element of a matrix ‘A’ 1. m = n by its complex conjugate is called the ‘conjugate matrix’ of 2. aij = 0 if i < j 1 0 0 0 A and is denoted by A. If A = [aij]m×n, then A =  aij  where  −2 aij is the conjugate of ‘aij’. 1 0 0 For example,   0 7 8 0 Properties of Conjugate of a Matrix 5 4 2 1   C − 1: (( A)) = Afor any matrix ‘A’ Horizontal Matrix If the number of rows of a matrix is less C − 2: ( A + B ) = A + B for any matrices A, B of same order. than the number of columns, i.e., m < n, then the matrix is C - 3: ( KA) = K A for any matrix ‘A’ and any Scalar K. called horizontal matrix. C − 4: ( AB) = ( A) ⋅ Bfor any matrices A and B with the con- Vertical Matrix If the number of columns in a matrix is dition that number of columns of A = number of less than the number of rows, i.e., if m > n, then the matrix rows of B. is called a vertical matrix. C − 5: ( A) n = ( An )for any square matrix ‘A’. Comparable Matrices Two matrices A = [aij]m×n and B = [bij]p×q are said to be comparable, if they are of same order, Tranjugate or Transposed Conjugate i.e., m = p; n = q. of a Matrix Equal Matrices Two comparable matrices are said to be Tranjugate of a matrix ‘A’ is obtained by transposing the ‘equal’, if the corresponding elements are equal, i.e., A conjugate of A and is denoted by Aq. Thus Aθ = ( A)T. = [aij]m×n and B = [bij]p×q are equal if Properties of Tranjugate of a Matrix 1. m = p; n = q (i.e., they are of the same order) TC - 1: (Aq )q = A for any matrix A 2. aij = bij ∀ i, j (i.e., the corresponding elements are TC - 2: (A + B)q = Aq + Bq for any matrices A, B of the equal) same order. TC - 3: (KA)q = KAq for any matrix A and any scalar K. Transpose of a Matrix TC - 4: (BA)q = BqAq for any matrix A, B with the condi- tion that number of columns of A = number of The matrix obtained by interchanging the rows and the col- rows of B. umns of a given matrix ‘A’ is called the ‘transpose’ of A and is denoted by AT or A′. If A is an (m × n) matrix, AT will TC - 5: (An)q = (Aq)n for any square matrix ‘A’. be an (n × m) matrix. Thus if A = [aij]m×n then AT = [uij]n×m, Symmetric Matrix A matrix A is said to be symmetric, if AT where uij = aji. = A (i.e., transpose of A = A). Chapter 04.indd 72 5/19/2017 5:19:16 PM  Chapter 4 Linear Algebra | 2.73 NOTE S - 3: a(bA) = (ab)A A symmetric matrix must be a square matrix. S - 4: 1A = A Skew-symmetric Matrix A matrix ‘A’ is said to be skew- Addition of Matrices symmetric matrix, if AT = (-A), i.e., A = [aij]m×n is skew sym- If A and B are two matrices of the same order, then they are metric if ‘conformable’ for addition and their sum ‘A + B’ is obtained 1. m = n by adding the corresponding elements of A and B, i.e., if 2. ajI = - aij ∀ i, j A = [aij]m×n; B = [bij]m×n, then A + B = [aij + bij]m×n. NOTE Properties of Addition Let A, B and C be three matrices of same order say m × n, then In a skew-symmetric matrix, all the elements of the prin- cipal diagonal are zero. A - 1: A + B is also a m × n matrix (CLOSURE) A - 2: (A + B) + C = A + (B + C) (ASSOCIATIVITY) Orthogonal Matrix A square matrix ‘A’ of order n × n is A - 3: If ‘O’ is the m × n zero (null) matrix, then A + O = O said to be an orthogonal matrix, if AAT = ATA = In. + A = A (‘O’ is the ADDITIVE IDENTITY) Involutory Matrix A square matrix ‘A’ is said to be involu- A - 4: A + (-A) = (-A) + A = O (-A is the ADDITIVE tory matrix, if A2 = I (where I is identity matrix). INVERSE) Idempotent Matrix A square matrix ‘A’ is said to be an A - 5: A + B = B + A (COMMUTATIVITY) idempotent matrix, if A2 = A. Nilpotent Matrix A square matrix ‘A’ is said to be nilpotent NOTE matrix, if there exists a natural number ‘n’ such that An = O. The set of matrices of same order form an ‘Abelian Group’ If ‘n’ is the least natural number such that An = O, then ‘n’ under addition. is called the index of the nilpotent matrix ‘A’. (Where O is the null matrix). Multiplication of Matrices Unitary Matrix A square matrix ‘A’ is said to be a unitary Let A and B be two matrices. A and B are conformable for matrix if, AAq = AqA = I. (Where Aq is the transposed con- multiplication, only if the number of columns of A is equal jugate of A.) to the number of rows of B. Hermitian Matrix A matrix ‘A’ is said to be a hermitian Let A = [aij] be an m × n matrix, B = [bjk] be an n × p matrix, if Aq = A, i.e., A = [aij]m×n is hermitian if matrix. Then the product ‘AB’ is defined as the matrix C = [cik] of order m × p where cik = ai1b1k + ai 2 b2 k +  + ain bnk 1. m = n n 2. aij = aij ∀i, j = ∑ aij b jk. j =1 NOTE cij calculated for i = 1, 2,... m and k = 1, 2,..., p will give The diagonal elements in a hermitian matrix are real numbers. all the elements of the matrix C. Skew-hermitian Matrix A matrix ‘A’ is said to be a skew- Properties of Multiplication hermitian matrix, if Aq = -A. M - 1: If A, B, C be m × n, n × p, p × q matrices respec- tively, then (AB)C = A(BC) (ASSOCIATIVITY). Operations on Matrices M - 2: If A is a m × n matrix, then A In = A and Im A = A and if A is a square matrix, i.e., m = n, then AI = Scalar Multiplication of Matrices IA = A (I is the MULTIPLICATIVE IDENTITY). If A is a matrix of order m × n and ‘K’ be any scalar (a M - 3: If A, B, C be m × n, n × p, p × q matrices respectively, real or complex number), then KA is defined to be a m × then A(B + C) = AB + AC (DISTRIBUTIVE LAW). n matrix whose elements are obtained by multiplying each M - 4: Matrix multiplication is NOT COMMUTATIVE element of ‘A’ by K, i.e., if A = [aij]m×n then KA = [Kaij]m×n in in general. particular if K = -1; then KA = -A is called the negative of M - 5: The INVERSE of a given matrix may not always exist. A and is such that, A + (-A) = [aij] + [-aij] = [aij - aij] = = O (zero matrix) Determinants (-A) + A = [-aij] + [aij] = [-aij + aij] = = O Let A = [aij] be a square matrix of order ‘n’. Then the deter- That is, A + (-A) = (-A) + A = O. minant of order ‘n’ associated with ‘A’ is denoted by | A | or Properties of Scalar Multiplication |aij| or Det(A) or D. Let A, B are two matrices of same order and a, b are any NOTES scalars, then S - 1: a(A + B) = aA + aB 1. Determinant of a matrix exists, only if it is a square matrix. S - 2: (a + b)A = aA + bA 2. The value of a determinant is a single number. Chapter 04.indd 73 5/19/2017 5:19:16 PM  2.74 | Part II Engineering Mathematics Determinant of Order 1 (or First a1 b1 c1 Order Determinant) Let ∆ = a2 b2 c2 If ‘a’ be any number, then determinant of ‘a’ is of order ‘1’ a3 b3 c3 and is denoted by |a|. The value of |a| = a. Enter the first column and then the second column after the Determinant of Order 2 (or Second Order third column and take the product of numbers as shown by Determinant) the arrows, taking care of signs indicated a1 b1 c1 a1 b1 If ‘A’ is a square matrix of order 2 given by a b  a1 b1 a2 b2 c2 a2 b2 A =  1 1  then Det ( A) = is determinant of  a2 b2  a2 b2 a3 b3 c3 a3 b3 order 2 and its value is D = a1b2 - a2b1 Then Minor and Cofactor of a Matrix D = a1b2c3 + b1c2a3 + c1a2b3 - a3b2c1 - b3c2a1 - c3a2b1  a1 b1 c1  We can now define the cofactor of an element aij in a 4 × 4   Let A =  a2 b2 c2  be a 3 × 3 matrix matrix as (-1)i+j × (Determinant of the 3 × 3 matrix obtained a b3 c3  by deleting the i-th row and j-th column) and determinant of  3 a 4 × 4 matrix to be the sum of products of elements of any Then the minor of an element aij of ‘A’ is the determinant of row (or column) by their corresponding cofactors. We can the 2 × 2 matrix obtained after deleting the i-th row and j-th similarly define determinant of a square matrix of any order. column of A and is denoted by Mij. Properties of Determinant The cofactor of aij is denoted by Aij and is defined as (-1)i+j Mij, i.e., Aij = (-1)i+j Mij 1. If two rows (or columns) of a determinant are interchanged, the value of the determinant is multiplied Determinant of Order 3 (Third by (-1). Order Determinant) 2. If the rows and columns of a determinant are If A is a square matrix of order ‘3’, given by interchanged, the value of the determinant remains  a1 b1 c1  unchanged, i.e., Det(A) = Det(AT).   3. If all the elements of a row (or column) of a A =  a2 b2 c2 . Then the determinant of ‘A’ is given by a determinant are multiplied by a scalar (say ‘K’), the  3 b3 c3  value of the new determinant is equal to ‘K’ times the a1 b1 c1 value of the original determinant. ∆ = Det A = a2 b2 c2 is a determinant of order 3 and 4. If two rows (or columns) of a determinant are identical, then the value of the determinant is zero. a3 b3 c3 5. If the elements of a row (or a column) in a determinant the value is obtained by taking the sum of the products of are proportional to the elements of any other row (or the elements of any row (or column) by their corresponding column), then the determinant is ‘0’. cofactors. Thus for A, D = a1A1 + b1B1 + c1C1 6. If every element of any row (or column) is zero, then determinant is ‘0’.    b2 c2 a2 c2 a2 b2 7. If each element in a row (or column) of a determinant is = a1 − b1 + c1 b3 c3 a3 c3 a3 b3 the sum of two terms, then its determinant can be or also D = a1A1 + a2A2 + a3A3 expressed as the sum of two determinants of the same order. b2 c2 b1 c1 b c1 = a1 − a2 + a3 1 8. (The theorem of ‘false cofactor’) The sum of products    b3 c3 b3 c3 b2 c2 of elements of a row (or column) with the cofactors of (This is by expanding by C1) and so on. any other row (or column) is zero. The sign to be used before a particular element can be  a1 b1 c1  judged by using the following rule:   Thus in A =  a2 b2 c2  +-+ a b c   3 3 3 -+- a1A2 + b1B2 + c1C2 = 0 +-+ The value of the determinants of order 3 can also be evalu- a2A1 + b2B1 + c2C1 = 0 and so on in general ated by using ‘Sarrus’ method given as follows: arAs + brBs + crCs = 0 if r ≠ s Chapter 04.indd 74 5/19/2017 5:19:17 PM  Chapter 4 Linear Algebra | 2.75 9. If the elements of a determinant are polynomials in x Results and the determinant vanishes for x = a, then x - a is a 1. If ‘A’ is of order 3 × 3 and K is any number, then factor of the determinant. Adj(KA) = K ²(Adj A). 2. A(Adj A) = (Adj A)A = |A| I for any square matrix ‘A’. Singular and Non-singular Matrices 3. Adj I = I; Adj O = O where I is the identity matrix and A square matrix ‘A’ is said to be singular, if Det(A) = 0 and O is the null matrix. is non-singular, if Det(A) ≠ 0. 4. Adj(AB) = (Adj B) (Adj A) if A, B are non-singular and are of same type. NOTES 5. If A = An ×n, then 1. A unit matrix is non-singular (since its Det = 1) 2. If A and B are non-singular matrices of the same det(Adj A) = (det A)n–1. ‘type’, then AB is non-singular of the same ‘type’. Adj(Adj A) = (det A)n–2(A). 2 |Adj(Adj A)| = (det A)(n–1) Inverse of a Matrix Evaluating Inverse of a Square Matrix Let ‘A’ be a square matrix. A matrix ‘B’ is said to be an inverse of ‘A’, if AB = BA = I. 1 If A is a square matrix, then A−1 = ( Adj A) A NOTE NOTES If B is the inverse of ‘A’, then ‘A’ is the inverse of ‘B’. 1. The inverse of an identity matrix is itself. 1 Some Results of Inverse 2. ( Adj A) −1 = A A 1. Inverse of a square matrix, when it exists, is unique. 3. If A is a non-singular square matrix (say of order 3) 2. The inverse of a square matrix exists, if and only if it and K is any non-zero number, then is non-singular. 1 −1 3. If ‘A’ and ‘B’ are square matrices of the same order, ( KA) −1 = A then ‘AB’ is invertible (i.e., inverse of AB exists) if ‘A’ K and ‘B’ are both invertible. 4. If ‘A’ and ‘B’ are invertible matrices of the same Rank and Nullity of a Matrix order, then (AB)-1 = B-1 A-1. Rank of a Matrix The Matrix ‘A’ is said to be of rank ‘r’, if 5. If A is invertible, then so is AT and (AT)-1 = (A-1)T. and only if it has at least one non-singular square sub-matrix of order ‘r’ and all square sub-matrices of order (r + 1) and 6. If A is invertible, then so is Aq and (Aq)-1 = (A-1)q. higher orders are singular. The rank of a matrix A is denoted by rank (A) or r(A). Adjoint of a Matrix Nullity of a Matrix If A is a square matrix of order ‘n’, then The adjoint of a square matrix ‘A’ is the transpose of the n - r(A), i.e., n - rank (A) is defined as nullity of matrix ‘A’ matrix obtained by replacing the elements of ‘A’ by their and is denoted by N(A). corresponding cofactors. Remark 1: If there is a non-singular square sub-matrix of NOTE order ‘K’, then r(A) ≥ K. The adjoint is defined only for square matrices and Remark 2: If there is no non-singular square sub-matrix of the adjoint of a matrix ‘A’ is denoted by Adj(A). If order ‘K’, then r(A) < K.  a1 a2  an  Remark 3: If A′ is the transpose of A, then r(A) = r(A′).   Remark 4: The rank of a null matrix is ‘0’. b1 b2  bn  A=      Remark 5: The rank of a non-singular square matrix of   order ‘n’ is ‘n’ and its nullity is ‘0’.  l1 l2  l n  T Remark 6: Elementary operations do not change the rank  A1 A2  An   A1 B1  L1  of a matrix.     B1 B2  Bn   A2 B2  L2  Remark 7: If the product of two matrices A and B is Adj A =  =             defined, then r(AB) ≤ r(A) and r(AB) ≤ r(B). That is, the     rank of product of two matrices cannot exceed the rank of  L1 L2  Ln   An Bn  Ln  either of them. Chapter 04.indd 75 5/19/2017 5:19:18 PM  2.76 | Part II Engineering Mathematics Elementary Operations or Elementary 1 3 2  Transformations Example: B = 3 4 −4  1. Elementary row operations 1 1 6  (a) Ri ↔ Rj: Interchanging of ith and jth rows (b) Ri → KRi: Multiplication of every element of ith 1 0 1  1 row with a non-zero scalar K C2 − 3C1 , C3 ∼ 3 −5 −2  = C (say ) 2 (c) Ri → Ri + kRj: Addition of k times the elements of 1 −2 6  jth row to the corresponding elements of ith row. C is a column equivalent to B. 2. Elementary column operations (a) Ci ↔ Cj: Interchanging of ith and jth columns Row Reduced Matrix A matrix A of order m × n is said to be row reduced if, (b) Ci → KCi: Multiplication of every element of ith column with a non-zero scalar K. 1. The first non-zero element of a non-zero row is 1. (c) Ci → Ci + KCj: Addition of K times the elements 2. Every other element in the column in which such 1’s of jth column to the corresponding elements of occur is 0. ith column.  2 3 −4 1  1 0 2 Example: Consider the matrix A =  3 0 1 5    A =  0 1 3  is a row reduced matrix  4 7 1 2  0 0 0   2 3 − 4 1  1 0 4     R2 → 2R2 ∼ 6 0 2 10  B =  0 5 0  is not a row reduced matrix.  4 7 1 2  0 0 0   2 − 4 3 1 Row Reduced Echelon Matrix A matrix ‘X ’ is said to be C 2 ↔ C3 ∼  3 1 0 5  row reduced echelon matrix if,    4 1 7 2  1. X is row reduced. 2. There exists integer P(0 ≤ p ≤ m) such that first ‘p’  0 − 4 3 1 rows of X are non-zero and all the remaining rows are C1 → C1 - 2C4 ∼  −7 1 0 5  zero rows.  0 1 7 2  3. For the ith non-zero row, if the first non-zero element NOTE of the row (i.e., 1) occurs in the jth column then, j1 < The rank of a matrix is invariant under elementary operations j2 < j3 <... < jp. 1 0 2 0 Row and Column Equivalence Matrices   0 1 2 0 0 1 0 3   Example: P =  ; Q = 0 0 0 1  Row Equivalence Matrix If B is a matrix obtained by 0 0 4 1   0 0 0 0 applying a finite number of elementary row operations   successively on matrix A, then matrix B is said to be row 0 0 0 0 equivalent to A (or a row equivalent matrix of A). are echelon matrices. The number of non-zero rows (i.e., value of P and Q) are 3 and 2 respectively. The value of i and Column Equivalence Matrix If B is obtained by applying a j are tabulated below finite number of elementary column operations successively i 1 2 3 i 1 2 on matrix A, then matrix B is said to be column equivalent P: Q: to A (or a column equivalent matrix of A ). j 1 2 3 j 2 4    1 3 4  Normal form of a Matrix Example: A =  2 5 −2  By means of elementary transformations, every matrix ‘A’ 1 4 −3 of order m × n and rank r (> 0) can be reduced to one of the following forms. 1 3 4   I 0 Ir  R2 − 2 R1 , R3 − R1 ∼ 0 −1 −10  = B (say )  1.  r    2. [Ir/0]  3. [Ir]  4.  0   0 0   0 1 −7  and these are called the normal forms. Ir is the unit matrix B is a row equivalent matrix of A. of order ‘r’. Chapter 04.indd 76 5/19/2017 5:19:19 PM  Chapter 4 Linear Algebra | 2.77 NOTE When the system of equations has one or more solutions, the equations are said to be CONSISTENT and the system If a m × n matrix ‘A’ has been reduced to the normal form of equations are said to be INCONSISTENT if it does not  I 0 say  r  then ‘r’ is the rank of A. admit any solution. The system of equations (1) is said to be  0 0 HOMOGENEOUS, if B = 0 NON-HOMOGENEOUS, if B ≠ 0 Let the system of equations be Systems of Linear Equations a11x1 + a12x2 +... + a1nxn = b1 Let a11 x1 + a12 x2 +  + a1n X n = b1   a12x1 + a22x2 +... + a2nxn = b2 a21 x1 + a22 x2 +  + a2 n X n = b2 ................................... ................................ ...  am1x1 + am2x2 +... + amnxn = bm...   This is a system of ‘m’ equations in ‘n’ variables x1, x2,... , an1 x1 + an 2 x2 +  ann xn = bn  (1) xn. The system of equations can be written as AX = B where be a system of ‘n’ linear equations in ‘n’ variables x1, x2,... , xn. The above system of equations can be written as  a11 a12  a1n   x1   b1         a21 a22  a2 n   x2  b2  a11 a12  a1n   x1   b1  A= ,X = , B=                         a21 a22  a2 n   x2  =  b2  or AX = B  am1 am 2 amn   xn   bm                  an1 an 2  ann   xn   bn   a11 a12  a1n b1    a a  a2 n b2  where The matrix  21 22 is called the augmented         a11 a12  a1n   x1   b1   am1 am 2  amn bm        a21 a22  a2 n  x2 b2 matrix of the system of equations and is denoted by [A : B]. A= , X =  , B =            Let AX = B represents ‘m’ linear equations with ‘n’       variables. Let rank of A = r and rank (A, B) = r1 [where (A,  an1 an 2  ann   xn   bn  B) is an augmented matrix]. If r1 ≠ r, then the system of A is called the co-efficient matrix. equations are inconsistent. If r1 = r, the table follows: Any set of values of x1, x2, x3,... which simultaneously satisfy these equations is called a solution of the system. m=n m>n m