AI Matrix Calculator Powered by Gemini

Perform matrix operations with detailed step-by-step solutions. Calculate determinants, inverses, eigenvalues, and more with AI-powered explanations.

Calculate Matrices Learn More
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a₁₁ a₁₂
a₂₁ a₂₂
]

Perform matrix operations:

Addition
Multiplication
Determinant
Inverse

Matrix Operations Calculator

Choose an operation, enter your matrices, and get detailed step-by-step solutions.

Select Matrix Operation:

Enter Matrix Dimensions:

Rows:
Columns:

Matrix A:

Try These Examples:

Matrix Addition

2x2 matrices

Matrix Multiplication

2x2 matrices

Determinant

2x2 matrix

Inverse Matrix

2x2 matrix

Key Features

Comprehensive Operations

Perform all major matrix operations: addition, multiplication, determinant, inverse, transpose, eigenvalues, rank, and solving linear systems.

Visual Matrix Display

Visualize matrices with heat maps and 3D plots. See matrix transformations and understand the geometric interpretation of matrix operations.

AI-Powered Explanations

Get detailed step-by-step solutions with clear explanations of each matrix operation. Learn the underlying mathematical concepts and algorithms.

Frequently Asked Questions

What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental mathematical objects used to represent linear transformations, systems of linear equations, and various mathematical structures. An m × n matrix has m rows and n columns.

What are the basic matrix operations?

The basic matrix operations include:

  • Addition: Adding corresponding elements (matrices must have same dimensions)
  • Scalar Multiplication: Multiplying every element by a scalar
  • Matrix Multiplication: Dot product of rows and columns (columns of first must equal rows of second)
  • Transpose: Flipping rows and columns
  • Determinant: A scalar value that can be computed from square matrices
  • Inverse: A matrix that when multiplied with original gives identity matrix

What is the determinant used for?

The determinant provides important information about a square matrix:

  • If determinant = 0, the matrix is singular (no inverse exists)
  • Absolute value represents scaling factor of linear transformation
  • Sign indicates whether transformation preserves orientation
  • Used in solving systems of linear equations (Cramer's rule)
  • Used in calculating eigenvalues and eigenvectors
  • Geometrically, it represents area/volume scaling in 2D/3D transformations

What are eigenvalues and eigenvectors?

For a square matrix A, an eigenvector v is a non-zero vector that when multiplied by A only changes by a scalar factor λ (the eigenvalue). Formally: Av = λv. Eigenvalues and eigenvectors are fundamental in:

  • Principal Component Analysis (PCA) in data science
  • Vibration analysis in mechanical engineering
  • Quantum mechanics in physics
  • Stability analysis in control systems
  • Google's PageRank algorithm
  • Diagonalization of matrices

Can this calculator handle complex matrices?

Yes, our matrix calculator can handle various types of matrices:

  • Real number matrices (most common)
  • Large matrices up to 6×6 in the web interface
  • Square matrices for determinant, inverse, eigenvalues
  • Rectangular matrices for multiplication (with compatible dimensions)
  • Sparse matrices (with many zeros)
  • Symmetric, diagonal, and triangular matrices

The AI provides detailed explanations for all operations, helping you understand both the computation process and the mathematical concepts behind matrix operations.

How to Use This Tool

1

Choose Matrix Operation

Select what you want to calculate: addition, multiplication, determinant, inverse, transpose, eigenvalues, rank, or solving a system of equations.

2

Enter Matrix Dimensions

Set the number of rows and columns for your matrix/matrices. The interface will automatically generate the appropriate input fields.

3

Fill Matrix Values

Enter the values for each matrix element. Use the example buttons to quickly load common matrix problems.

4

Calculate & Explore Results

Click "Calculate & Explain" to get the result along with step-by-step solutions, matrix properties, and visualizations.

Pro Tip

Use the quick example buttons to see different types of matrix operations. For matrix multiplication, make sure the number of columns in the first matrix equals the number of rows in the second matrix.