matrix-Matrix Calculations Online

Example

Given two matrices A and B of the same dimensions, their sum or difference is obtained by adding or subtracting corresponding elements. For instance, if A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then A + B = [[6, 8], [10, 12]].

Scenario

Matrix addition and subtraction are often used in image processing where pixel intensities need to be adjusted.

Example

Matrix multiplication involves taking the dot product of rows and columns. For example, if A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then A * B = [[19, 22], [43, 50]].

Scenario

Matrix multiplication is crucial in computer graphics for transformations such as scaling, rotating, and translating objects.

Example

The determinant of a square matrix provides information about the matrix, such as whether it is invertible. For example, the determinant of A = [[1, 2], [3, 4]] is -2.

Scenario

In engineering, determinants are used to solve systems of linear equations and analyze structural stability.

Example

The inverse of a square matrix A is a matrix B such that A * B = I, where I is the identity matrix. For example, the inverse of A = [[4, 7], [2, 6]] is B = [[0.6, -0.7], [-0.2, 0.4]].

Scenario

Matrix inverses are used in cryptography for encryption and decryption processes.

Example

Eigenvalues and eigenvectors of a matrix A are scalars and vectors such that A * v = λ * v, where λ is an eigenvalue and v is an eigenvector. For example, for A = [[1, 2], [4, 3]], one eigenvalue is 5 with eigenvector [2, 4].

Scenario

In physics, eigenvalues and eigenvectors are used to analyze vibrational modes and quantum mechanics problems.