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 c_{ij}= a_{ij}+ b_{ij}

## C++

`// 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++

`// 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++

`// 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

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

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