## Objective

Implement two functions in PHP to return the sum and product of two given matrices.

Easy

## Description

A matrix is a rectangular block of nubmers comprised of rows and columns. A matrix with $m$ rows and $n$ columns is said to be an $m \times n$ matrix.

Shown below are a couple of matrices, along with their dimensions:

$\begin{bmatrix} 1 & 5 \\ 10 & -2 \end{bmatrix}$
$2 \times 2$
$\begin{bmatrix} 1 & 3 \\ 0 & 0 \\ 6 & 15 \end{bmatrix}$
$3 \times 2$
$\begin{bmatrix} 1 & 9 & 0 & -8 & 10 & 0 \end{bmatrix}$
$1 \times 6$

In PHP, a matrix can be represented using a 2D array, as follows:

<?php

$matrix = [ [1, 5], [10, -2] ];$matrix = [
[1, 3],
[0, 0],
[6, 15]
];

$matrix = [ [1, 9, 0, -8, 10, 0] ]; Each row is represented as an array inside the main array. Then inside this inner array, each element represents an entry of the given row. Matrices can be added, subtracted, and multiplied together. The addition is represented as $\text{A} + \text{B}$ and is defined if and only if $\text{A}$ is $m \times n$ and $\text{B}$ is also $m \times n$. The addition is computed by adding corresponding items in $\text{A}$ and $\text{B}$ together. Following is an illustration: $\begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 5 & -3 \\ 6 & 10 \end{bmatrix} = \begin{bmatrix} 1 + 5 & 3 + (-3) \\ 0 + 6 & 0 + 10 \end{bmatrix} = \begin{bmatrix} 6 & 0 \\ 6 & 10 \end{bmatrix}$ The case for multiplication is quite different. The multiplication of $\text{A}$ and $\text{B}$ is denoted as $\text{AB}$, and is defined if the number of columns of $\text{A}$ is the same as the number of rows of $\text{B}$. In other terms, $\text{AB}$ is defined if and only if $\text{A}$ is $m \times p$ and $\text{B}$ is $p \times n$. It is computed as follows. The $(i, j)^{th}$ element of $\text{AB}$ is computed by multiplying corresponding elements in the $i^{th}$ row of $\text{A}$ and the $j^{th}$ column of $\text{B}$ and then adding them together. An example is shown below: $\begin{bmatrix} 1 & 2 \end{bmatrix} \times \begin{bmatrix} 0 & 3 \\ 1 & 5 \end{bmatrix} = \begin{bmatrix} (1 \times 0 + 2 \times 1) & (1 \times 3 + 2 \times 5) \end{bmatrix} = \begin{bmatrix} 2 & 13 \end{bmatrix}$ In this exercise, you have to create two functions multiply and add() that work as follows: 1. add() takes in two matrices $a and $b as arguments and returns their sum, as another matrix. 2. multiply() takes in two matrices $a and $b as arguments and returns their product, as a matrix. For the sake of brevity and simplicity, you shall assume that the arguments provided to add() and multiply() are valid, i.e. you don't have to worry about implementing the functions so as to detect any invalid matrices passed in. In a later exercise in the upcoming units, we'll extend both these functions with error-handling capabilities in them. Anyways, shown below is an illustration of the functions in action: <?php /* Functions defined here */$a = [ [1, 3], [0, 0] ];
$b = [ [5, -3], [6, 10] ]; print_r(add($a, $b));$a = [ [1, 2] ];
$b = [ [0, 3], [1, 5] ]; print_r(multiply($a, \$b));
Array (  => Array (  => 6  => 0 )  => Array (  => 6  => 10 ) ) Array (  => Array (  => 2  => 13 ) )