Matrices: Overview | Mathematics (Maths) Class 12 - JEE

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:
  Abbreviated as: A =  [a_ij]_{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 a_ij belongs to the i^th row and j^th column and is called the (i, j)^th element of the matrix A = [a_ij].

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, ...... a_1n] 
2. Column Matrix: Matrix having one column i.e. matrix of order m × 1. They are also known as column vectors.
   Example: 
3. Zero or Null Matrix: An m × n  matrix all whose entries are zero. It is denoted as O_m×n.
   Example:
4. Horizontal Matrix: A matrix of order m × n is a horizontal matrix if n > m.
   Example:
5. Verical Matrix: A matrix of order m × n is a vertical matrix if m > n.
   Example: 
6. Square Matrix: If number of rows is equal to number of column, then the matrix is a square matrix.
   Example:

   Note:

   In a square matrix,

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

   (b) The elements  a₁₁, a₂₂, a₃₃, ...... a_nn 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 a_ii is known as trace of the matrix. It is denoted as t_r(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, ....., d_n).
   Example:

   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, a_ij = 0, for i ≠ j and a_ij = k, for i = j
   Example:
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, a_ij = 0, for i ≠ j and a_ij = 1, for i = j
   Example:

Equality of Matrices 

A = [a_ij] & B = [b_ij] will be equal, only if

• Both have the same order
• a_{i j} = b_{i j}  for each pair of i & j

Operations on Matrices 

For the Matrices,

• Addition of Matrices: A +  B = [ a_ij + b_ij], where  A & B are of the same order.
  i.e.
  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., 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
  
  i.e.
  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 , I^m  =  I  for  all  m ∈ N.

Illustration 1. Find the value of x and y, if 
Ans.

∴
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 , 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.

 

BA is not possible since the number of columns of B ≠ number of rows of A.

Matrix Polynomial

• If f(x) = a₀xⁿ + a₁x^{n – 1} + a₂x^{n – 2} + ......... + a_nx⁰, then we define a matrix polynomial f(A) = a₀Aⁿ + a₁A^{n– 1} + a₂A^n–2 + ..... + a_nIⁿ 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 A^m = O, A^m–1 ≠ O

(c) Periodic Matrix: A square matrix is periodic when it satisfies the relation A^K+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.

Transpose of a Matrix  

• The transpose of a matrix is obtained by changing its rows & columns.
• It is denoted as A^T or  A′.
• If a matrix be A =  [ a_ij ] of order m x n, then  A^T or  A′ =  [ a_ji ]  for 1 ≤ i ≤ n & 1 ≤ j ≤ m of order n × m
  i.e.
• Exxample: 
• Reversal law for transpose: (A₁, A₂, ...... A_n)^T =
  

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

Symmetric & Skew-Symmetric Matrix 

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

Note: 

(i) For symmetric matrix, A = A^T

(ii) Max. number of distinct entries in a symmetric matrix of order n is

(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 A^T = A and A is skew-symmetric, if A^T = − A.
(b) A + A^T is a symmetric matrix and A − A^T is a skew symmetric matrix.
Consider  (A + A^T)^T =  A^T  + (A^T)^T =  A^T + A   =  A + A^T. Thus, A + A^T is symmetric.
Similarly, we can prove that A − A^T is 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.

Adjoint of A Square Matrix

Let  be a square matrix and let the matrix formed by the cofactors of [a_{i j} ] in determinant A is =  Then (adj A) =

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| I_n, 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) = K^n–1 (adj A), K is a scalar

Illustration 3. If , find adj A.
Ans. Each element of cofactor matrix

Cofactor matrix =
adj A = transpose of codaftor matrix =

Elementary Transformation

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

• Interchanging any two rows (or columns), denoted by R_i↔R_j or C_i↔C_j
  
• Multiplication of the element of any row (or column) by a non-zero quantity and denoted by Ri → kRi or Ci → kCj
  
• 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
  

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|  I_n
  A⁻¹ A (adj A) =  A⁻¹ I_n |Α|
  I_n (adj A)  =  A⁻¹ |A|  I_n 
  ∴ A⁻¹  =

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 A^T is also invertible & (A^T)⁻¹ = (A⁻¹)^T.
• If A is invertible,
  (a) (A⁻¹)⁻¹ = A  ;  
  (b) (A^k)⁻¹ =  (A⁻¹)^k = A^–k, k ∈ N 
• If A is an Orthogonal Matrix. AA^T = I = A^TA
• A square matrix is said to be orthogonal if ,  A⁻¹ = A^T.
• 

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:

i.e.

Let
⇒ AX = B 
⇒ A⁻¹ A X  =  A⁻¹ B   
⇒ X  =  A⁻¹ B = 

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
   

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:


∵X  =  A⁻¹ B = 
Therefore,


⇒