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 B_m*n gives a matrix C_m*n. The elements of C are sum of corresponding elements in A and B which can be shown as:

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 A_m*n and B_m*n gives a matrix C_m*n. The elements of C are difference of corresponding elements in A and B which can be represented as:

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 A_m*n and B_n*p gives a matrix C_m*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:

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