Matrices & Determinants: Matrix Algebra - 1 | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Matrices - Introduction

A matrix A over a field K or, simply, a matrix A (when K is implicit) is a rectangular array of scalars usually presented in the following form:

• The rows of such a matrix A are the m horizontal lists of scalars: (a₁₁, a₁₂, ..., a_1n), (a₂₁, a₂₂, ..., a_2n), ..., (a_m1, a_m2, ..., a_mn).
  and the columns of A are the n vertical lists of scalars:
  
• Note that the element a_ij, called the ij-entry or ij-element, appears in row i and column j.
  We frequently denote such a matrix by simply writing A = [a_ij].
• A matrix with m rows and n columns is called an m by n matrix, written m * n. The pair of numbers m and n is called the size of the matrix. 
• Two matrices A and B are equal, written A = B, if they have the same size and if corresponding elements are equal. Thus the equality of two m * n matrices is equivalent to a system of mn equalities, one for each corresponding pair of elements.
• A matrix with only one row is called a row matrix or row vector and a matrix with only one column is called a column matrix or column vector. 
• A matrix whose entries are all zero is called a zero matrix and will usually be denoted by 0.
• Matrices whose entries are all real numbers are called real matrices and are said to be matrices over R. 
• Analogously, matrices whose entries are all complex numbers are called complex matrices and are said to be matrices over ℂ.

Example:

1. The rectangular array  is a 2 x 3 matrix. Its rows are (1, - 4 , 5)  and (0,3, 2), and its columns are
2. The 2 x 4 zero matrix is the matrix .

Algebra of matrix

Matrix Addition and Scalar Multiplication

Let A = [ a_ij and B = [b_ij] be two matrices with the same size , say m * n matrices. The sum of A and B, written A + B, is the matrix obtained by adding corresponding elements from A and B. That is,

The product of the matrix A by a scalar k, written k • A or simply kA, is the matrix obtained by multiplying each element of A by k. That is,

Observe that A + B and kA are also m * n matrices. We also define-A= (-1) A and A - B = A + (-B)
The matrix - A is called the negative of matrix A, and the matrix A - B is called the difference of A and B. The sum of matrices with different sizes is not defined.

Example:

The matrix 2A - 3B is called a linear combination of A and B.
The basic properties of matrices under the operations of matrix addition and scalar multiplication are as follows:
Consider any matrices A, B, and C (with the same size) and any scalars k and k’. Then:
(i) (A + B) + C = A + (B + C ),
(ii) A + 0 = 0 + A = A,
(iii) A + (- A) = (- A) + A = 0,
(iv) A + B = B + A,
(v) k(A + B) = kA + kB,
(vi) (k + k’)A = kA + k’A,
(vii) (k k’)A = k (k’A),
(viii) 1· A = A.

Matrix Multiplication

The product of matrices A and B, written AB, is somewhat complicated. For this reason, we first begin with a special case.
The product AB of a row matrix A = [a_n] and a column matrix B = [b_n] with the same number of elements is defined to be the scalar (or 1 * 1 matrix) obtained by multiplying corresponding entries and adding; that is,

We emphasize that AB is a scalar (or a 1 x 1 matrix). Product AB is not defined when A and B have different numbers of elements.
Example of product AB of a row matrix and a column matrix:

Method of multiplication of two matrices

Suppose A = [a_ik] and B = [b_kj] are matrices such that the number of columns of A is equal to the number of rows of B; say, A is an m * p matrix and B is a p * n matrix. Then the product AB is the m x n matrix whose ij-entry is obtained by multiplying the ith row of A by the jth column of B. That is,

where

The product AB is not defined if A is an m * p matrix and B is a q * n matrix.
Theorem: Let A, B, and C be matrices. Then, whenever the products and sums are defined:
(i) (AB)C = A(BC) (associative law),
(ii) A(B + C) = AB + AC (left distributive law),
(iii) (B + C)A = BA + CA (right distributive law),
(iv) k(AB) = (kA)B = A(kB), where k is a scalar.
We note that OA = 0 and BO = 0, where 0 is the zero matrix.

Transpose of a Matrix

The transpose of a matrix A, written A^T, is the matrix obtained by writing the columns of A, in order, as rows. For example,

• In other words, if A = [a_ij] is an m * n matrix, then A^T = [b_ij] is the n * m matrix where b_ij = a_ji.
• Observe that the transpose of a row vector is a column vector. 
• Similarly, the transpose of a column vector is a row vector.

The next theorem lists the basic properties of the transpose operation.
Theorem: Let A and B be matrices and let k be a scalar. Then, whenever the sum and product are defined:
(i) (A + B)^T = A^T + B^T,
(ii) (A^T)^T = A,
(iii) (kA)^T = kAT,
(iv) (AB)^T = B^TA^T.

Square Matrices

A square matrix is a matrix with the same number of rows as columns. An n * n square matrix is said to be of order n and is sometimes called an n-square matrix.

Example: The following are square matrices of order 3.

The following are also matrices of order 3:

Question for Matrices & Determinants: Matrix Algebra - 1

Try yourself: The matrix  is a

• A.

  Identity matrix
• B.

  Symmetric matrix
• C.

  Skew symmetric matrix
• D.

  None of these

Explanation

Let the given matrix be:

A = 
Let us find the transpose of A.

Therefore, A is a symmetric matrix.

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Diagonal and Trace

Let A = [ a_ij] be an n-square matrix. The diagonal or main diagonal of A consists of the elements with the same subscripts, that is, a₁₁, a₂₂, a₃₃, ... a_nn.
The trace of A, written tr(A), is the sum of the diagonal elements. Namely, tr(A) = a₁₁+ a₂₂+ a₃₃+ ...+ a_nn
The following theorem applies.
Theorem: Suppose A = [a_ij] and B = [b_ij] are n-square matrices and k is a scalar. Then:
(i) tr(A + B) = tr(A) + tr(B),
(ii) tr(kA) = k tr(A),
(iii) tr(AT) = tr(A),
(iv) tr(AB) = tr(BA).

Example:
Let

Diagonal of A = {1, -4, 7} and tr(A) = 1 - 4 + 7 = 4. Diagonal of B = {2, 3, -4} and tr(B) = 2 + 3 - 4 = 1
Moreover,
tr(A + B ) = 3 - 1 + 3 = 5.
tr(2A) = 2 - 8 + 14 = 8.
tr(A^T) = 1 -4 + 7 = 4
tr(AB) = 5 + 0 - 35 = -30,
tr(BA) = 27 - 24 - 33 = -30
As expected from Theorem,
tr(A + B) = tr(A) + tr(B),
tr(A^T) = tr(A), tr(2A) = 2 tr(A)  
Furthermore, although AB * BA, the trace are equal.

Power of Matrices, Polynomials in Matrices

Let A be an n-square matrix over a field K. Powers of A are defined as follows:
A² = AA,  A³ = A²A,  ... , Aⁿ⁺¹ = Aⁿ A, ....,  and A⁰ = I
Polynomials in matrix A are also defined. Specifically, for any polynomial
f(x) = a₀ + a₁x + a₂x² +...+ a_nxⁿ
where the a_i are scalars in K, f(A) is defined to be the following matrix:
f(x) = a₀I + a₁A + a₂A² +...+ a_nAⁿ

Note - f(A) is obtained from f(x) by substituting the matrix A for the variable x and substituting the scalar matrix a₀l for the scalar a₀. If f(A) is the zero matrix, then A is called a zero or root of f(x).

Example: Suppose  Then

Suppose f(x) = 2x² - 3x + 5 and g(x) = x² + 3x - 10. Then


Thus A is a zero of the polynomial g(x).

Types of Matrices

• Row Matrix: If a matrix has only one row and any number of columns, is called a Row matrix, or a row vector. Example - [2 7 3 9] is a row matrix.
• Column Matrix: A matrix having one column and any number of rows is called a column matrix or a column vector. Example:  is a column matrix.
• Null Matrix or Zero Matrix: Any matrix in which all the elements are zeros is called a Zero matrix or Null matrix, i.e., . In general, an m * n matrix in which each element is zero is called a null matrix of the type m * n and is denoted by O_m*n.
• Square Matrix: A matrix in which the number of rows is equal to the number of columns is called a square matrix i.e. A = (a_ij)_m*n is a square matrix if and only if m=n.
    are square matrices.
• Diagonal Matrix: A square matrix is called a diagonal matrix if all its non-diagonal elements are zero i.e. in general a matrix A = (a_ij)_mxn is called a diagonal matrix if a_ij = 0 for i ≠ j;
  for example  is a diagonal matrix of order 3 * 3.
• Unit or Identity Matrix: A square matrix is called a unit matrix if all the diagonal elements are unity and non-diagonal elements are zero.
   are Identity matrices of order 3 x 3 and 2 * 2 respectively.
  The identity matrix I is similar to the scalar 1 in that, for any n-square matrix A,
  Al = IA = A.More generally, if B is an m x n matrix, then Bl_n = l_mB = B.
• Symmetric Matrix: A square matrix is called symmetric if for all values of i and j, a_ij = a_ji .
  ⇒ A’ = A i.e. transpose of the matrix A = A ;
   is a symmetric matrix of order 3 * 3.
• Skew Symmetric Matrix: A square matrix is called a skew-symmetric matrix if a_ij = -a_ji, for all values of i and j and all its diagonal elements are zero.
  ⇒ A’ = - A i.e. transpose of the matrix A = -A;
  example.  is a skew-symmetric matrix of order 3 x 3.
• Triangular Matrix: A square matrix, in which all its elements below the leading diagonal are zero, is called an upper triangular matrix while a square matrix, all of whose elements above the leading diagonal are zero, is called a lower triangular matrix.
  
• Transpose of Matrix: As discussed above if in a given matrix, we interchange the rows and the corresponding columns, the new matrix obtained is called the transpose of the matrix A and is denoted by A or A^t or A^T.
  
• Orthogonal Matrix: A square matrix A is called an orthogonal matrix if the product of matrix A with its transpose matrix A’ is an identity matrix.
  i.e. AA’ = I.
  
  Question for Matrices & Determinants: Matrix Algebra - 1

  Try yourself:The matrix  is a

  • A.

    Identity Matrix
  • B.

    Symmetric Matrix
  • C.

    Skew Symmetric Matrix
  • D.

    None of these

  Explanation

  

  Report a problem

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  View Solution
• Conjugate of a Matrix: A conjugate matrix of a matrix is obtained by replacing each term with its complex conjugate. Complex conjugate means that the sign of the imaginary part of all its complex numbers has been changed. The Conjugate of a matrix is denoted by the bar over the name of the matrix as Ā.
  
  
  ∴ Conjugate of matrix
  
• Conjugate Transpose of a Matrix, A^θ: Transpose of the conjugate of a matrix A is denoted by A^θ.
  we haveand
• Unitary Matrix: A square matrix A is said to be unitary if we have A^θA =1
  Remarks
  • A real matrix is unitary if and only if it is orthogonal.
  • If P is unitary so  are also unitary.
  • If P and Q are unitary so PQ is unitary multiple.
  • If P is unitary, then I P I  is of unit modulus.
  • Any two Eigen vectors corresponding to the distinct Eigen values of a unitary matrix are orthogonal.
  • If P is orthogonal so P^T and P^-1 are also orthogonal.
  • If P and Q are orthogonal so PQ is also orthogonal.
  • If P is orthogonal, then IPI = ± 1.
• Hermitian Matrix: A square matrix A = (a_ij) is said to be Hermitian Matrix if every i - jth element of A is equal to the conjugate complex of j - ith element of A, or in order words
  The necessary and sufficient condition for the matrix A to be Hermitian is that A = A^θ i.e. conjugate transpose of .
• Skew Hermitian Matrix: A square A = (a_ij) is said to be a Skew Hermitian matrix if every i-jth element of A is equal to the negative conjugate complex of j-ith element of A; i.e  All the elements of the principal diagonal are of the form.
  Hence a_ij is purely imaginary i.e. a_ij = 0 + ib or a_ij = 0.
  i.e. all the diagonal element of a Skew Hermitian Matrix are either zeros or pure imaginary.
  
  Thus the necessary and sufficient condition for a matrix A to be Skew Hermitian is that
  Properties
  • If A is a Skew-Hermitian matrix, then kA is also Skew-Hermitian for any real number k.
  • If A and B are Skew-Hermitian matrices of same order, then λ1A +λ₂B is also Skew Hermitian for real number λ1,λ₂.
  • If A and B are Hermitian matrices of same order, then AB-BA is Skew-Hermitian.
  • If A is any square matrix, then A - A* is Skew-Hermitian matrices.
  • Every square matrix can be uniquely represented as the sum of a Hermitian and a Skew Hermitian matrices.
  • If A is a skew-Hermitian (Hermitian) matrix, then iA is a Hermitian matrix, then  also Skew-Hermitian (Hermitian).
• Idempotent Matrix: A matrix such that A² = A is called Idempotent Matrix.
  
• Periodic Matrix: A matrix A is said to be a Periodic Matrix if A^k+1 = A, where k is a +ve integer. If k is the least +ve integer for which A^{k +1} = A, then k is said to be the period of A. If we choose k = 1, then A² = A and we call it to be an idempotent matrix.
• Involutory Matrix: A matrix A is said to an involutory matrix if A² = I (unit matrix). Since A² = I always for unit matrix, we conclude that the unit matrix is always involutory.
• Nilpotent Matrix: A matrix is said to be a Nilpotent matrix if A^k = 0 (null matrix) where k is +ve integer; if however k is the least +ve integer of which A^k = 0, then k is called the index of the nilpotent matrix.
  The matrix  is nilpotent and its index is 2.
• Submatrix: A matrix obtained from a given matrix, say A = (a_ij)_mxn by deleting some rows or columns or both is called a submatrix of A.
• Principal diagonal: The elements a₁₁, a₃₃, ..., a_mn of a square matrix of order m*m are called diagonal elements and the diagonal containing these elements is called the principal diagonal.  In a square matrix (a_ij)_{m x m}, a_ij is the diagonal element if i = j.
• Scalar Matrix: For any scalar k, the matrix kl that contains k’s on the diagonal and 0's elsewhere is called the scalar matrix corresponding to the scalar k. Observe that
  (kl)A = k(IA) = kA. That is, multiplying a matrix A by the scalar matrix kl is equivalent to multiplying A by the scalar k.
  Example : The following are the identity matrices of orders 3 and 4 and the corresponding scalar matrices for k = 5;
  
  Remark 1: It is common practice to omit blocks or patterns of 0's when there is no ambiguity, as in the above second and fourth matrices.
  Remark 2: The Kronecker delta function δ_ij is defined by
  Thus the identity matrix may be defined by I = [δ_ij]

Equal Matrices

Two matrices are equal if 

1. They are of the same order.
2. The elements in the corresponding positions are equal.

Thus if  Here A = B.

In general if A = (a_ij)_{m x d} and B = (b_ij)_{m x n} are matrices each of order m * n and a_ij = b_ij for all i and j then A = B.

Note: 

1. An orthogonal matrix P is said to be proper if IPI = 1, improper if IPI = - 1.
   Clearly, P^-1 is proper of both improper, the PQ is proper if PQ is improper then one of P, Q is improper.
2. A square matrix is unitary if and only if its columns (rows) form an orthogonal set of unit vectors.
3. A square matrix is orthogonal if and only if its columns (rows) form an orthogonal set of unit vectors.
4. Let x, be any unit n-vector. Then there exists a unitary matrix U having x₁ as its first column.