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Definition

  • A set of numbers or objects or symbols represented in the form of a rectangular array is called a matrix. 
  • The order of the matrix is defined by the number of rows and number of columns present in the rectangular array of representation. 
  • Unlike determinants, it has no value.
  • A matrix of order m × n, i.e. m rows and n columns, is represented below:
    Overview: Matrices | Mathematics (Maths) for JEE Main & AdvancedAbbreviated as: A =  [aij]m × n, where 1 ≤ i ≤ m; 1 ≤ j ≤ n, i denotes the row and j denotes the column.
  •  The number a11, a12, ….. etc., are known as the elements of matrix A, where aij belongs to the ith row and jth column and is called the (i, j)th element of the matrix A = [aij].

Type of Matrices

  1. Row Matrix: Matrix having one row i.e. matrix of order 1 × n. They are also known as row vectors.
    Example: A =  [a11 , a12, ...... a1n
  2. Column Matrix: Matrix having one column i.e. matrix of order m × 1. They are also known as column vectors.
    Example:
     Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  3. Zero or Null Matrix: An m × n  matrix all whose entries are zero. It is denoted as Om×n.
    Example: Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  4. Horizontal MatrixA matrix of order m × n is a horizontal matrix if n > m.
    Example: Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  5. Verical Matrix: A matrix of order m × n is a vertical matrix if m > n.
    Example: Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  6. Square Matrix: If number of rows is equal to number of column, then the matrix is a square matrix.
    Example: Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

    Note:

    In a square matrix, Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

    (a) The pair of elements aij & aji are called Conjugate Elements.
    Example: a12 and a21

    (b) The elements  a11, a22, a33, ...... ann are called Diagonal Elements. The line along which the diagonal elements lie is called the "Principal or Leading" diagonal.

    (c) Sum of all the diagonal elements, i.e. Σ aii is known as trace of the matrix. It is denoted as tr(A).

  7. Diagonal Matrix: A square matrix in which all the elements are zero except the diagonal element. It is denoted as dia(d1, d2, ....., dn).
    Example: Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

    Note:

    Min. number of zeros in a diagonal matrix of order n = n(n – 1)

  8. Scalar Matrix: A square matrix in which every non-diagonal element is zero and all diagonal elements are equal, is called a scalar matrix.
    i.e. in scalar matrix, aij = 0, for i ≠ j and aij = k, for i = j
    Example:Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  9. Unit/Identity Matrix: A square matrix, in which every non-diagonal element is zero and every diagonal element is 1, is called, unit matrix or an identity matrix.
    i.e. in scalar matrix, aij = 0, for i ≠ j and aij = 1, for i = j
    Example:Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Equality of Matrices 

A = [aij] & B = [bij] will be equal, only if

  • Both have the same order
  • ai j = bi j  for each pair of i & j

Question for Overview: Matrices
Try yourself:
Which type of matrix has every non-diagonal element as zero and all diagonal elements as equal?
View Solution

Operations on Matrices 

For the Matrices,Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

  • Addition of Matrices: A +  B = [ aij + bij], where  A & B are of the same order.
    i.e.Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
    Properties:
    (a) Addition of matrices is commutative i.e. A + B  =  B + A.
    (b) Matrix addition is associative i.e. (A + B) + C =  A + (B + C)
    (c) Additive inverse. If A + B = O = B + A , then A and B are additive inverse of each other.
  • Multiplication of a Matrix by a Scalar: If a matrix is multiplied by a scalar quantity, then each element is multiplied by that quantity for the resulting matrix.
    i.e.Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced, where k is a scalar quantity.
  • Multiplication of Matrices: Two matrices A, B can be multiplied to give resulting matrix AB, only if, no. of a column of A(prefactor) is equal to the no. of rows of B (post factor)
    i.e. A is a matrix of order n x m and B is a matrix of order p x q, then AB exists only if m = p.
    If m=p, order of AB = n x q
    Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
    i.e.Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
    Properties:
    (a) Matrix multiplication may or maynot be commutative i.e. AB may or may not be equal to BA.
    (b) Matrix multiplication is Associative i.e. (A . B) . C  =  A . (B . C)
    (c) Matrix multiplication is Distributive over Matrix Addition i.e. A(B+C) = AB + AC
    (d) Cancellation Laws not necessary hold in case of matrix multiplication i.e. if AB = AC ⇒ B = C even if A ≠ 0.
    (e) AB = 0 i.e., Null Matrix, does not necessarily imply that either A or B is a null matrix.
    (f) Positive Integral Powers of a Square Matrix i.e. For a square matrix A , A2A = (AA) A  = A (AA) = A3

Note: 

  • If A and B are two non-zero matrices such that AB = O then A and B are called the divisors of zero.
  •  If A and B are two matrices such that:
    (i) AB = BA ⇒ A and B commute each other
    (ii) AB = – BA ⇒ A and B anti commute each other
  • For a unit matrix I of any order , Im  =  I  for  all  m ∈ N.

Illustration 1. Find the value of x and y, if Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Ans. Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
On equating the corresponding elements of L.H.S. and R.H.S.

A2 + y = 5 ⇒ y = 3
2x + 2 = 8 ⇒ 2x = 6 ⇒ x = 3

Thus, x = 3 and y = 3.


Illustration 2. If Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced, find AB and BA if possible.
Ans. A is a 3 × 3 matrix and B is a 3 × 2 matrix, therefore, A and B are conformable for the product AB and it is of the order 3 × 2.

Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced 
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
BA is not possible since the number of columns of B ≠ number of rows of A.

Question for Overview: Matrices
Try yourself:
Which of the following properties hold true for matrix multiplication?
View Solution
 

Matrix Polynomial

  • If f(x) = a0xn + a1xn – 1 + a2xn – 2 + ......... + anx0, then we define a matrix polynomial f(A) = a0An + a1An– 1 + a2An–2 + ..... + anIn where A is the given square matrix.
  • If f (A) is the null matrix then A is called the zero or root of the polynomial f (x).

Special Matrices

(a) Idempotent Matrix: A square matrix is idempotent provided  A2 = A

(b) Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order m, m ∈ N , if Am = O, Am–1 ≠ O

(c) Periodic Matrix: A square matrix is periodic when it satisfies the relation AK+1 = A, for some positive integer K. The period of the matrix is the least value of K for which this holds true.

(d) Involutary Matrix: If A2 = I , the matrix is said to be an involutary matrix.

Note:

  • For an idempotent matrix, An = A ∀ n > 2, n ∈ N.
  • Period of an idempotent matrix is 1.
  • For an involuntary Matrix, A = A–1.

Question for Overview: Matrices
Try yourself:
Which of the following properties is true for the transpose of a matrix?
View Solution

Transpose of a Matrix  

  • The transpose of a matrix is obtained by changing its rows & columns.
  • It is denoted as AT or  A′.
  • If a matrix be A =  [ aij ] of order m x n, then  AT or  A′ =  [ aji ]  for 1 ≤ i ≤ n & 1 ≤ j ≤ m of order n × m
    i.e. Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  • Exxample: Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  • Reversal law for transpose: (A1, A2, ...... An)T =Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

Properties of Transpose:
If AT &  Bdenote the transpose of A and B.
(a) (A ± B)T =  AT  ± BT ;  note that  A & B  have the same order.
(b) (A B)T  =  BT  AT A & B  are conformable for matrix product AB.
(c) (AT)T = A
(d) (kA)T = k AT k is a scalar.

Symmetric & Skew-Symmetric Matrix 

  • A square matrix A = [ aij]  is said to be, symmetric if , aij = aji ∀ i & j (conjugate  elements are equal) 
  • A square matrix A = [ aij]  is said to be, Skew-symmetric if , aij = -aji ∀ i & j  (the pair of conjugate elements are additive inverse of each other)

Note: 

(i) For symmetric matrix, A = AT

(ii) Max. number of distinct entries in a symmetric matrix of order n is Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

(iii) The digaonal elements of a skew symmetric matrix are all zero, but not the converse i.e. if digaonal elements are 0 doesn't mean matrix is skew symmetric.

Properties Of Symmetric & Skew Matrix

(a) A is symmetric, if AT = A and A is skew-symmetric, if AT = − A.
(b) A + AT is a symmetric matrix and A − AT is a skew symmetric matrix.
Consider  (A + AT)T =  AT  + (AT)T =  AT + A   =  A + AT. Thus, A + AT is symmetric.
Similarly, we can prove that A − Ais skew-symmetric.
(c) The sum of two symmetric matrices is a symmetric matrix and the sum of two skew-symmetric matrices is a skew-symmetric matrix.
(d) If  A & B  are symmetric matrices then,
(AB + BA)  is a symmetric matrix and (AB − BA) is a skew-symmetric matrix.
(e) Every square matrix can be uniquely expressed as a sum of a symmetric and a skew-symmetric matrix.
i.e.Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

Question for Overview: Matrices
Try yourself:
If matrix A is given as , what is the adjoint of matrix A?
View Solution
 

Adjoint of A Square Matrix

Let Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced be a square matrix and let the matrix formed by the cofactors of [ai j ] in determinant A is = Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced Then (adj A) = Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

Note: Co-factors of the elements of any matrix are obtained by eliminating all the elements of the same row and column and calculating the determinant of the remaining elements.

Theorem :  A (adj. A) = (adj. A).A = |A| In, if A be a square matrix of order n. 

Properties:

(i) | adj A | = | A |n – 1 

(ii) adj (AB) = (adj B) (adj A) 

(iii) adj(KA) = Kn–1 (adj A), K is a scalar

Illustration 3. If Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced, find adj A.
Ans. Each element of cofactor matrix
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Cofactor matrix = Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
adj A = transpose of codaftor matrix = Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

[Intext Question]

Elementary Transformation

Any one of the following operations on a matrix is called an elementary transformation.

  • Interchanging any two rows (or columns), denoted by Ri↔Rj or Ci↔Cj
    Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  • Multiplication of the element of any row (or column) by a non-zero quantity and denoted by Ri → kRi or Ci → kCj
    Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
  • Addition of constant multiple of the elements of any row to the corresponding element of any other row, denoted by Ri → Ri + kRj or Ci → Ci + kCj
    Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

Inverse Of A Matrix (Reciprocal Matrix) 

  • A square matrix A said to be invertible (non singular) if there exists a matrix B such that, AB = I = BA. B is called the inverse (reciprocal) of A and is denoted by A−1
  • Thus A−1 =  B  ⇔  AB = I = BA. 
  • We have, A.(adj A)  =  |A|  In
    A−1 A (adj A) =  A−1 In |Α|
    In (adj A)  =  A−1 |A|  In 
    ∴ A−1  = Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

Imp. Theorem:  If A & B are invertible matrices of the same order , then  (AB)−1  = B−1A−1. This is reversal law for inverse

Note: 

  • The necessary and sufficient condition for a square matrix A to be invertible is that |A| ≠ 0.
  • If A be an invertible matrix, then AT is also invertible & (AT)−1 = (A−1)T.
  • If A is invertible,
    (a) (A−1)−1 = A  ;  
    (b) (Ak)−1 =  (A−1)k = A–k, k ∈ N 
  • If A is an Orthogonal Matrix. AAT = I = ATA
  • A square matrix is said to be orthogonal if ,  A−1 = AT.
  • Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

System of Equation & Criterian for Consistency Gauss - Jordan Method

a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3

We can write it in the form of matrix: Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

i.e. Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

Let Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
⇒ AX = B 
⇒ A−1 A X  =  A−1 B   
⇒ X  =  A−1 B = Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

Note:

  • If |A| ≠ 0, system is consistent having unique solution
  • If  |A| ≠ 0 & (adj A). B  ≠ O (Null matrix), system is consistent having unique non − trivial solution.
  • If |A| ≠ 0 & (adj A) . B = O (Null matrix) , system is consistent having trivial solution
  •  If |A| = 0, matrix method fails
    Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced 

Illustration 4. Solve the following equation,
2x+y+2z=0, 2x-y+z=10, x+3y-z=5.

Ans. The given equation can be written in matrix form:Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
X  =  A−1 B = Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Therefore,
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced
Overview: Matrices | Mathematics (Maths) for JEE Main & Advanced

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FAQs on Overview: Matrices - Mathematics (Maths) for JEE Main & Advanced

1. What are the different types of matrices?
Ans. Matrices can be classified into different types such as square matrices, rectangular matrices, diagonal matrices, identity matrices, zero matrices, symmetric matrices, skew-symmetric matrices, etc.
2. What operations can be performed on matrices?
Ans. Operations that can be performed on matrices include addition, subtraction, scalar multiplication, matrix multiplication, finding the determinant, finding the inverse, and finding the transpose.
3. How is the transpose of a matrix defined?
Ans. The transpose of a matrix is obtained by interchanging rows and columns of the original matrix. It is denoted by A^T, where A is the original matrix.
4. What is the significance of the inverse of a matrix?
Ans. The inverse of a matrix is important in solving systems of linear equations, as it helps in finding the solution efficiently. A matrix A has an inverse if and only if its determinant is non-zero.
5. What are elementary transformations in matrices?
Ans. Elementary transformations in matrices involve operations such as swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row. These transformations are used in row reduction to find the echelon form or reduced row-echelon form of a matrix.
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