---
id: matrix
title: Matrix
toc_max_heading_level: 2
---

## Introduction

A matrix is a 2-dimensional array. Questions involving matrices are usually related to [dynamic programming](./dynamic-programming.md) or [graph](./graph.md) traversal.

Matrices can be used to represent graphs where each node is a cell on the matrix which has 4 neighbors (except those cells on the edge and corners). This page will focus on questions which don't use matrix as graphs. Questions which are meant to use the matrix as a graph can be found on the [graph section](./graph.md).

## Corner cases

- Empty matrix. Check that none of the arrays are 0 length
- 1 x 1 matrix
- Matrix with only one row or column

## Techniques

### Creating an empty N x M matrix

For questions involving traversal or dynamic programming, you almost always want to make a copy of the matrix with the same size/dimensions that is initialized to empty values to store the visited state or dynamic programming table. Be familiar with such a routine in your language of choice:

This can be done easily in Python in one line.

```py
# Assumes that the matrix is is non-empty
zero_matrix = [[0 for _ in range(len(matrix[0]))] for _ in range(len(matrix))]
```

Copying a matrix in Python is:

```py
copied_matrix = [row[:] for row in matrix]
```

### Transposing a matrix

The transpose of a matrix is found by interchanging its rows into columns or columns into rows.

Many grid-based games can be modeled as a matrix, such as Tic-Tac-Toe, Sudoku, Crossword, Connect 4, Battleship, etc. It is not uncommon to be asked to verify the winning condition of the game. For games like Tic-Tac-Toe, Connect 4 and Crosswords, where verification has to be done vertically and horizontally, one trick is to write code to verify the matrix for the horizontal cells, transpose the matrix, and reuse the logic for horizontal verification to verify originally vertical cells (which are now horizontal).

Transposing a matrix in Python is simply:

```py
transposed_matrix = zip(*matrix)
```

## Recommended questions

- [Set Matrix Zeroes](https://leetcode.com/problems/set-matrix-zeroes/)
- [Spiral Matrix](https://leetcode.com/problems/spiral-matrix/)
- [Rotate Image](https://leetcode.com/problems/rotate-image/)
- [Valid Sudoku](https://leetcode.com/problems/valid-sudoku/)

## Recommended courses

import AlgorithmCourses from '../\_courses/AlgorithmCourses.md'

<AlgorithmCourses />