Different Operations on Matrices



For introduction on matrices, you can refer the following article: Matrix Introduction
In this article, we will discuss various operations on matrices and their properties:

Matrices Addition –
The addition of two matrices A m*n and Bm*n gives a matrix Cm*n. The elements of C are sum of corresponding elements in A and B which can be shown as:

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The algorithm for addition of matrices can be written as:

for i in 1 to m
   for j in 1 to n
      cij = aij + bij

Key points:



  • Addition of matrices is commutative which means A+B = B+A
  • Addition of matrices is associative which means A+(B+C) = (A+B)+C
  • The order of matrices A, B and A+B is always same
  • If order of A and B is different, A+B can’t be computed
  • The complexity of addition operation is O(m*n) where m*n is order of matrices

Matrices Subtraction –
The subtraction of two matrices Am*n and Bm*n gives a matrix Cm*n. The elements of C are difference of corresponding elements in A and B which can be represented as:

2

The algorithm for subtraction of matrices can be written as:

for i in 1 to m
   for j in 1 to n
      cij = aij-bij

Key points:

  • Subtraction of matrices is non-commutative which means A-B ≠ B-A
  • Subtraction of matrices is non-associative which means A-(B-C) ≠ (A-B)-C
  • The order of matrices A, B and A-B is always same
  • If order of A and B is different, A-B can’t be computed
  • The complexity of subtraction operation is O(m*n) where m*n is order of matrices

Matrices Multiplication –
The multiplication of two matrices Am*n and Bn*p gives a matrix Cm*p. It means number of columns in A must be equal to number of rows in B to calculate C=A*B. To calculate element c11, multiply elements of 1st row of A with 1st column of B and add them (5*1+6*4) which can be shown as:

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The algorithm for multiplication of matrices A with order m*n and B with order n*p can be written as:

for i in 1 to m
   for j in 1 to p
      cij = 0
for k in 1 to n
    cij += aik*bkj

Key points:

  • Multiplication of matrices is non-commutative which means A*B ≠ B*A
  • Multiplication of matrices is associative which means A*(B*C) = (A*B)*C
  • For computing A*B, the number of columns in A must be equal to number of rows in B
  • Existence of A*B does not imply existence of B*A
  • The complexity of multiplication operation (A*B) is O(m*n*p) where m*n and n*p are order of A and B respectively
  • The order of matrix C computed as A*B is O(m*p) where m*n and n*p are order of A and B respectively

Read next – Determinant of a Matrix, Adjoint and Inverse of a Matrix



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