Matrix Operations

Matrices

Addition and Subtraction of Matrices

• Two matrices can be added/subtracted if they have the same order (number of rows and columns)
• Addition/Subtraction is done element-wise, i.e., corresponding elements of the matrices are added/subtracted
• The resulting matrix has the same order as the original matrices

Matrices

Multiplication of Matrices

• Two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix
• The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix
• The elements of the resulting matrix are obtained by multiplying the corresponding elements of the rows of the first matrix and columns of the second matrix

Matrices

Inverse of a Matrix

• A matrix A has an inverse A⁻¹ if and only if it is a square matrix (number of rows = number of columns) and its determinant is non-zero
• The inverse of a matrix A satisfies the property: A × A⁻¹ = A⁻¹ × A = I, where I is the identity matrix
• The inverse of a matrix can be used to solve systems of linear equations

Networks

Adjacency Matrix

• An adjacency matrix is a square matrix used to represent a network or graph
• The elements of the matrix indicate the presence (1) or absence (0) of an edge between two nodes in the network
• The adjacency matrix is symmetric if the network is undirected

Networks

Shortest Path Algorithms

• Shortest path algorithms are used to find the minimum distance between two nodes in a network
• Common algorithms include:
  1. Dijkstra's algorithm
  2. Bellman-Ford algorithm
  3. Floyd-Warshall algorithm
• These algorithms are used in various applications, such as:
  • Finding the shortest path between two cities on a map
  • Optimizing network traffic flow
  • Scheduling tasks in a project management system

Matrices

Addition and Subtraction of Matrices

• Matrices can be added or subtracted if they have the same order (number of rows and columns)
• Addition and subtraction are performed element-wise, corresponding elements are added or subtracted
• The resulting matrix has the same order as the original matrices

Multiplication of Matrices

• Matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix
• The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix
• Elements of the resulting matrix are obtained by multiplying corresponding elements of rows of the first matrix and columns of the second matrix

Inverse of a Matrix

• A matrix A has an inverse A⁻¹ if it is a square matrix and its determinant is non-zero
• The inverse of a matrix A satisfies the property: A × A⁻¹ = A⁻¹ × A = I, where I is the identity matrix
• The inverse of a matrix can be used to solve systems of linear equations

Networks

Adjacency Matrix

• An adjacency matrix is a square matrix used to represent a network or graph
• Elements of the matrix indicate the presence (1) or absence (0) of an edge between two nodes in the network
• The adjacency matrix is symmetric if the network is undirected

Shortest Path Algorithms

• Shortest path algorithms are used to find the minimum distance between two nodes in a network
• Dijkstra's algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm are common algorithms used
• Applications of shortest path algorithms include:
  • Finding the shortest path between two cities on a map
  • Optimizing network traffic flow
  • Scheduling tasks in a project management system