Matrix addition is a basic C++ procedure that merges two matrices to create a new matrix. Matrices are two-dimensional numerical arrays with rows and columns. The final matrix's members are generated by adding the corresponding elements of the two matrices.

We use placed loops to run through the rows and columns of the matrices to implement matrix addition in C++. We access the elements of each matrix within the loops, add them together, and store the results in a new matrix.

Matrix addition is frequently utilized in computer graphics, scientific computing, and data analysis. It helps the combining of data from several matrices as well as matrices' mathematical operations. Understanding matrix addition in C++ establishes foundations for more complex matrix operations and calculations.

## Understanding Matrices and their Representation

Matrices are mathematical structures that are made up of a rectangular array of elements that are organized in rows and columns. They are frequently utilized in various domains, including mathematics, computer science, physics, and data analysis.

Each element in a matrix is recognized by its location, which is given by a row index and a column index. The number of rows and columns knows the size of a matrix. For example, A matrix with m rows and n columns is considered m x n in size.

Matrixes can be represented in various ways depending on the computer language or mathematical notation. In computer languages like C++, arrays are widely used to represent matrices.

In a two-dimensional array format, each matrix row is saved as a separate array. After that, the arrays are joined together to form the matrix. To access the items, two indices are used: 1 for the row and 1 for the column.

For example, consider a matrix A with three rows and four columns:

In C++, this matrix can be represented using a two-dimensional array as follows:

Code

The matrix elements are stored in a single contiguous block of memory. The first index denotes the row, while the second denotes the column.

Sparse matrices in which only non-zero elements and their indices are saved, and linked lists, in which each element is represented by a node carrying its value, row index, and column index, are two more matrix representations. These representations are useful for efficiently storing and manipulating huge matrices with many zero entries. Understanding matrix representation is critical for performing addition, subtraction, multiplication, and inversion operations. It allows us to access and alter individual items and conduct actions on full rows or columns.

### Algorithm for Matrix Addition in C++

Here is a C++ algorithm for matrix addition:

1. Begin by defining two matrices with the same dimensions. Let's call them Matrix A and Matrix B.
2. To store the result of the addition, create a new matrix Matrix C with the same dimensions as Matrix A and Matrix B.
3. Use nested loops to iterate through each element of the matrices, with the outer loop iterating over the rows and the inner loop iterating over the columns.
4. Access the matching items from Matrix A and Matrix B at the current row and column location.
5. Add the elements' values from Matrix A and Matrix B
6. Put the total in the relevant row and column location of Matrix C.
7. Steps 4-6 should be repeated for all items in the matrix, ensuring the iterations cover the complete range of rows and columns.
8. Matrix C will hold the result of the addition of Matrix A and Matrix B when the iterations are completed.
9. Matrix C can then be used for other processes or shown as the appropriate result.

This procedure in C++ allows you to add matching elements from two matrices and store the result in a new matrix.

### Implementation in C++

The output of the above program will be:

```Matrix A:
1 2 3
4 5 6
7 8 9
Matrix B:
9 8 7
6 5 4
3 2 1
10 10 10
10 10 10
10 10 10
```

Explanation:

In this version, the matrix addition function performs matrix addition by iterating through the elements of two matrices, A and B, and storing the sum in another matrix C. The display matrix function is used to display the matrices.

In the main function, we define two matrices, matrixA, and matrixB, with their respective values. We also define matrix C to contain the result of the addition.

The matrix addition function is then applied to the matrices matrixA, matrixB, and matrixC, as well as the amount of rows and columns supplied. The existing matrices matrixA, matrixB, and the newly added matrixC matrix are then displayed using the display matrix function.

When you start this program, it will perform matrix addition and display the original and resultant matrices.

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