Matrix Data Structure is a two-dimensional array arranged in rows and columns. It is commonly used to represent mathematical matrices and is fundamental in various fields like mathematics, computer graphics, and data processing. Matrices allow for efficient storage and manipulation of data in a structured format.
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What is Matrix Data Structure?
Matrix is a two-dimensional array or table consisting of rows and columns. The intersection of a row and column is called a cell. All the data is stored across different cells in the matrix. Matrix data structure is used when we want to store data in the form of table or grid. Each element in a matrix is identified by its row and column indices.
Features of Matrix Data Structure:
- Size: A matrix has a specific size, defined by its number of rows and columns.
- Element: A matrix’s row and column indices serve to identify each entry, which is referred to as an element.
- Operations: Scalar multiplication and the operations of addition, subtraction, and multiplication on matrices are also supported.
- Determinant: A square matrix’s determinant is a scalar number that may be used to solve systems of linear equations and carry out other linear algebraic operations.
- Inverse: If a square matrix has an inverse, it may be used to solve linear equation systems and carry out other linear algebraic operations.
- Transpose: By flipping a matrix along its main diagonal and switching the rows and columns, you may create the transpose of the matrix.
- Rank: In many applications, including the solution of linear equations and linear regression analysis, the rank of a matrix—a measure of its linearly independent rows or columns—is utilized.
Applications of Matrix Data Structure:
- Linear Algebra: Matrices are widely used in linear algebra, a branch of mathematics that deals with linear equations, vector spaces, and linear transformations. Matrices are used to represent linear equations and to solve systems of linear equations.
- Optimization: Matrices are used in optimization problems, such as linear programming, to represent the constraints and objective functions of the problem.
- Statistics: Matrices are used in statistics to represent data and to perform operations such as correlation and regression.
- Signal Processing: Matrices are used in signal processing to represent signals and to perform operations such as filtering and transformation.
- Network Analysis: Matrices are used in network analysis to represent graphs and to perform operations such as finding the shortest path between two nodes.
- Quantum Mechanics: Matrices are used in quantum mechanics to represent states and operations in quantum systems.
Basics of Matrix Data Structure:
- Introduction to Matrix or Grid – Data Structure and Algorithms Tutorial
- Row-wise vs column-wise traversal of matrix
- Applications of Matrices and Determinants
Basic Operations on Matrix Data Structure:
- Traverse a given Matrix using Recursion
- Rotate Matrix Elements
- Sort the given matrix
- Search element in a sorted matrix
- Program to find transpose of a matrix
- Adjoint and Inverse of a Matrix
- Determinant of a Matrix
Standard Easy Problems on Matrix Data Structure:
- Print matrix in zag-zag fashion
- Program for scalar multiplication of a matrix
- Print a given matrix in spiral form
- Find distinct elements common to all rows of a matrix
- Find unique elements in a matrix
- Find maximum element of each row in a matrix
- Shift matrix elements row-wise by k
- Swap major and minor diagonals of a square matrix
- Squares of Matrix Diagonal Elements
- Sum of middle row and column in Matrix
- Program to check idempotent matrix
- Program to check diagonal matrix and scalar matrix
- Program for Identity Matrix
- Mirror of matrix across diagonal
- Program for addition of two matrices
- Program for subtraction of matrices
Standard Medium Problems on Matrix Data Structure:
- Program for Conway’s Game Of Life
- Program to multiply two matrices
- Rotate a matrix by 90 degree without using any extra space
- Check if all rows of a matrix are circular rotations of each other
- Given a matrix of ‘O’ and ‘X’, find the largest subsquare surrounded by ‘X’
- Maximum size square sub-matrix with all 1s
- Count zeros in a row wise and column wise sorted matrix
- Queries in a Matrix
- Find pairs with given sum such that elements of pair are in different rows
- Find all permuted rows of a given row in a matrix
- Find number of transformation to make two Matrix Equal
- Inplace (Fixed space) M x N size matrix transpose
- Minimum flip required to make Binary Matrix symmetric
- Magic Square
Standard Hard Problems on Matrix Data Structure:
- Find the number of islands
- A Boolean Matrix Question
- Matrix Chain Multiplication
- Maximum size rectangle binary sub-matrix with all 1s
- Construct Ancestor Matrix from a Given Binary Tree
- Print K’th element in spiral form of matrix
- Find size of the largest ‘+’ formed by all ones in a binary matrix
- Shortest path in a Binary Maze
- Print maximum sum square sub-matrix of given size
- Validity of a given Tic-Tac-Toe board configuration
- Minimum Initial Points to Reach Destination
- Strassen’s Matrix Multiplication
- Maximum sum rectangle in a 2D matrix
- Program for Sudoku Generator
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