Matrices

Operations on Matrices


  • A matrix A = [aij]m × n is said to be horizontal matrix if m < n
  • A matrix A = [aij]m × n is said to be vertical matrix if m > n
  • A matrix A = [aij]m × n is said to be square matrix if m = n
  • A matrix A = [aij]m × n is said to be Row matrix if m = 1
  • A matrix A = [aij]m × n is said to be column matrix if n = 1
  • A matrix A = [aij]m × n is said to be rectangular matrix if m = n
  • A matrix A = [aij]m x n is said to be zero matrix if aij = 0 ∀ i and j
  • A matrix A = [aij]m x n is said to be unit matrix if aij = \tt \begin{cases}1 & for \ i = j\\0 & for \ i \neq j\end{cases}
    \tt I_{2}=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}
    \tt I_{3}=\begin{bmatrix}1 & 0&0 \\0 & 1&0\\0 & 0&1 \end{bmatrix}
    \tt I_{4}=\begin{bmatrix}1 & 0&0&0 \\0 & 1&0&0\\0 & 0&1&0\\0 & 0&0&1 \end{bmatrix}
  • A square matrix A = [aij]m × m is said to be diagonal matrix if aij = 0 for i ≠ j
            Ex : \tt A=\begin{bmatrix}3 & 0&0 \\0 & 4&0\\0 & 0&7 \end{bmatrix}
  • If A = diag (d1. d2, d3 ..... dn) then \tt A^{n}=diag\left(d_1^n,d_2^n,d_3^n,....d_n^n\right)
  • A square matrix A = [aij]m × m is said to be scalar matrix if aij = \tt \begin{cases}k & for \ i = j\\0 & for \ i \neq j\end{cases}
           Ex : \tt A=\begin{bmatrix}4 & 0&0 \\0 & 4&0 \\0&0&4 \end{bmatrix}
  • A square matrix A = [aij]m × m is said to be upper triangular matrix id aij = 0 for i > j
          Ex : \tt A=\begin{bmatrix}-2 & 3&5 \\0 & 5&9 \\0&0&6 \end{bmatrix}
  • A square matrix A = [aij]m × n is said to be lower triangular matrix if aij = 0 for i < j 
          Ex : \tt A=\begin{bmatrix}-5 & 0&0 \\3 & 1&0 \\2&5&9 \end{bmatrix}
  • Equality of matrices: Two matrices A and B are said be equal if 
        i) Order of A = Order of B 
       ii) The corresponding elements of A and B are equal
  • Trace of the matrix : The sum of the principal diagonal elements of the square matrix A is called "Trace of matrix A
         Ex :\tt A=\begin{bmatrix}a_{11} & a_{12}&a_{13}\\a_{21} & a_{22}&a_{23}\\a_{31} & a_{32}&a_{33} \end{bmatrix}
         Tr (A) = a11 + a22 + a33
  • Tr (KA) = K trace (A)
    Tr (A + B) = tr (A) + tr (B)
    Tr (A − B) = tr (A) − tr (B)
    Tr (AB) ≠ tr (A) · tr (B)
    Tr (AB) = Tr (BA)
    Tr (ABC) = Tr (BCA) = Tr (CAB) = Tr (BAC) = Tr (CAB) = Tr (ACB)
  • Idempotent matrix : A square matrix A is said to be idempotent if A2 = A i.e if AB = A and BA = B then A2 = A, B2 = B
  • Involutery matrix : A square matrix A is called involutery matrix if A2 = I

Part1: View the Topic in this video From 41:07 To 57:05

Part2: View the Topic in this video From 00:40 To 09:50

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  • Addition of Matrics :
    If A = [aij]m x n B = [bij]m x n  ⇒ A + B = [aij + bij]m x n 
    → A + B = B + A (commutative)
    → A + (B + C) = (A + B) + C (Associative)
    → A + O = O + A = A (O is the additive identity)
    → A + (−A) = −A + A = 0 (The additive inverse of A is −A)
    → A + B = A + C ⇒ B = C
    → A + B = C ⇒ A = C − B
  • Multiplication of Matrices : If A and B are the two matrices, then its product AB is defined if the number of columns in A is equal to number of rows in B. i.e,
    A = [aij]m x n B = [bij]n x p
    ⇒ AB = [cij]m x p
  • If \tt A=\begin{bmatrix}a_{11} & a_{12} \\a_{21} & a_{22} \end{bmatrix}B=\begin{bmatrix}b_{11} & b_{12} \\b_{21} & b_{22} \end{bmatrix}
    \tt AB=\begin{bmatrix}a_{11} & b_{11}&+&a_{12} & b_{21}&& a_{11} & b_{12}&+&a_{12} & b_{22} \\a_{21} & b_{11}&+&a_{22} &        b_{21}&& a_{21} & b_{12}&+&a_{22} & b_{22} \end{bmatrix}
  • → AB ≠ BA  (Not commutative)
    → A(BC) = (AB) C (Associative)
    → AI = IA = A (I is multiplicative identity)
    → AA−1 = A−1 A = I (The multiplicative inverse of A is A−1)
    → (Am)n = Amn
    → Am. An = Am+n
    → A(B + C) = AB + AC
    → K (AB) = (KA) B = A (KB) for any scalar k.

1. Properties of matrix addition : If A, B and C are matrices of same order, then
    (i)   A + B = B + A                       (Commutative law)
    (ii)  (A + B) + C = A + (B + C)     (Associative law)
    (iii)  A + O = O + A = A, where O is zero matrix which is additive identity of the matrix.
    (iv)  A + (−A) = 0 = (−A) + A, where (−A) is obtained by changing the sign of every element of A, which is additive inverse of the matrix.
    (v) A + B = A + C and B + A = C + A   then B = C     (Cancellation law)

2. Properties of scalar multiplication : If A, B are matrices of the same order and λ, μ are any two scalars then
   (i)  λ(A + B) = λA + λB
   (ii) (λ + μ)A = λA + μA
   (iii) λ(μA) = (λμA) = μ(λA)
   (iv) (−λA) = −(λA) = λ(−A)

3. Properties of matrix multiplication : If A, B and C are three matrices such that their product is defined, then
   (i)   AB ≠ BA,               (Generally not commutative)
   (ii)  (AB)C = A(BC),      (Associative law)
   (iii) IA = A = AI, where I is identity matrix for matrix multiplication.
   (iv) A(B + C) = AB + AC, (Distributive law)
   (v)  If AB = AC \nRightarrow B = C, (Cancellation law is not applicable)

4. Remember that if A and B are two matrices of the same order, then
    (i) (A + B)2 = A2 + B2 + AB + BA
    (ii) (A − B)2 = A2 + B2 − AB − BA
    (iii) (A − B)(A + B) = A2 − B2 + AB − BA
    (iv) (A + B)(A − B) = A2 − B2 − AB + BA
    (v) A(−B) = (−A)(B) = −AB