Different Operations on Matrices | Engineering Mathematics - Civil Engineering (CE) PDF Download

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
c_ij = a_ij + b_ij
// C++ Program for matrix addition
#include <iostream>
using namespace std;
int main()
{
int n = 2, m = 2;
int a[n][m] = { { 2, 5 }, { 1, 7 } };
int b[n][m] = { { 3, 7 }, { 2, 9 } };
int c[n][m];
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++) {
c[i][j] = a[i][j] + b[i][j];
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++)
cout << c[i][j] << " ";
cout << endl;
}
}

Output:
5 12
3 16

Time Complexity: O(n * m)

Auxiliary Space: O(n * m)

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
c_ij = a_ij-b_ij
// C++ Program for matrix substraction
#include <iostream>
using namespace std;
int main()
{
int n = 2, m = 2;
int a[n][m] = { { 2, 5 }, { 1, 7 } };
int b[n][m] = { { 3, 7 }, { 2, 9 } };
int c[n][m];
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++) {
c[i][j] = a[i][j] - b[i][j];
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++)
cout << c[i][j] << " ";
cout << endl;
}
}

Output:
-1 -2
-1 -2

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
c_ij = 0
for k in 1 to n
c_ij + = a_ik * b_kj
// C++ Program for matrix Multiplication
#include <iostream>
using namespace std;
int main()
{
int n = 2, m = 2;
int a[n][m] = { { 2, 5 }, { 1, 7 } };
int b[n][m] = { { 3, 7 }, { 2, 9 } };
int c[n][m];
int i, j, k;
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
{
c[i][j] = 0;
for (k = 0; k < n; k++)
c[i][j] += a[i][k] * b[k][j];
}
}
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
cout << c[i][j] << " ";
cout << endl;
}
}

Output:
16 59
17 70

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 m * p where m * n and n * p are order of A and B respectively.