Matrix Calculator

Professional matrix calculator for linear algebra operations with step-by-step solutions

Matrix A
[
]
Matrix B
[
]

Result Matrix

[
0
]
Matrix Result
Operation: Addition
Matrix Size: 3×3
Result Type: Matrix
Matrix A
[
]

Determinant Result

0 det(A)
Matrix Size: 3×3
Determinant: 0
Matrix Type: Regular
Invertible: No

Calculation Steps:

1. Enter matrix values to see calculation steps
Matrix A
[
]

Matrix Inverse

[
A⁻¹
]
A⁻¹ (Inverse Matrix)
Matrix Size: 3×3
Determinant: 0
Invertible: No
Method: Gauss-Jordan
Matrix A
[
]

Eigenvalues

λ₁, λ₂, λ₃ Eigenvalues
Matrix Size: 3×3
Eigenvalue 1: -
Eigenvalue 2: -
Eigenvalue 3: -
Characteristic Poly: det(A - λI)

How to Use the Matrix Calculator

Our comprehensive matrix calculator provides four powerful tools for linear algebra operations with step-by-step solutions:

🔢 Basic Matrix Operations

Perform fundamental matrix operations including addition, subtraction, multiplication, transpose, and scalar multiplication. Supports matrices up to 4×4 size with customizable dimensions. Features convenient fill options (random, identity, zero) for quick testing and examples.

🔍 Determinant Calculator

Calculate determinants of square matrices (2×2, 3×3, 4×4) with detailed step-by-step solutions. Shows cofactor expansion, rule of Sarrus for 3×3 matrices, and indicates whether the matrix is invertible. Essential for solving systems of linear equations and matrix analysis.

↩️ Matrix Inverse

Find the inverse of square matrices using Gauss-Jordan elimination method. Automatically checks if the matrix is invertible by calculating its determinant. Shows the complete inverse matrix with verification that A × A⁻¹ = I (identity matrix).

🎯 Eigenvalues & Eigenvectors

Calculate eigenvalues and eigenvectors for 2×2 and 3×3 matrices. Solves the characteristic equation det(A - λI) = 0 to find eigenvalues. Useful for understanding matrix transformations, stability analysis, and principal component analysis.

🌟 Key Features

• Visual Matrix Input: Interactive grid interface with mathematical notation
• Multiple Matrix Sizes: Support for 2×2, 3×3, and 4×4 matrices
• Step-by-Step Solutions: Detailed mathematical explanations
• Quick Fill Options: Random, identity, zero, and example matrices
• Professional Notation: Proper mathematical symbols and formatting
• Error Handling: Clear feedback for invalid operations
• Educational Value: Perfect for learning linear algebra concepts

All calculations follow standard linear algebra methods and provide educational insights into matrix theory and applications.

Frequently Asked Questions

How do you multiply two matrices?
Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix. The result is calculated by taking the dot product of rows from the first matrix with columns from the second matrix. For example, a 3×2 matrix can be multiplied by a 2×4 matrix, resulting in a 3×4 matrix.
What is a matrix determinant and why is it important?
The determinant is a scalar value calculated from a square matrix that provides important information about the matrix. If the determinant is zero, the matrix is singular (non-invertible). The determinant is used to solve systems of linear equations, calculate matrix inverses, and determine if vectors are linearly independent.
When does a matrix have an inverse?
A square matrix has an inverse if and only if its determinant is non-zero. Such matrices are called invertible or non-singular. The inverse matrix A⁻¹ satisfies the property A × A⁻¹ = A⁻¹ × A = I (identity matrix). Not all matrices have inverses, only square matrices with non-zero determinants.
What are eigenvalues and eigenvectors used for?
Eigenvalues and eigenvectors describe how a matrix transformation affects certain vectors. An eigenvector only changes in magnitude (not direction) when multiplied by the matrix, and the eigenvalue is that scaling factor. They're used in principal component analysis, stability analysis, quantum mechanics, and many engineering applications.
Can you add matrices of different sizes?
No, matrix addition and subtraction require matrices to have exactly the same dimensions. You can only add or subtract matrices that have the same number of rows and columns. For example, you can add two 3×2 matrices, but you cannot add a 3×2 matrix to a 2×3 matrix.