Matrix Calculator

Perform comprehensive matrix operations including addition, multiplication, determinant, inverse, and more

Matrix Operations

Select operation:

Matrix dimensions:

Rows (Matrix A):

Columns (Matrix A):

Rows (Matrix B):

Columns (Matrix B):

Matrix A:

Matrix B:

Display options:

Show calculation steps Show as fractions when possible Round decimals to 4 places

Results

Configure matrices and click "Calculate" to see the results

Understanding Matrices

What is a Matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are fundamental in linear algebra and have numerous applications in mathematics, physics, engineering, and computer science.

Matrix Notation

A = [aᵢⱼ] where i = row, j = column

[a₁₁ a₁₂]
[a₂₁ a₂₂]

Matrix Properties

Matrices have various properties and classifications based on their dimensions, elements, and mathematical characteristics.

Common Types

• Square: Equal rows and columns
• Identity: Square with 1s on diagonal
• Zero: All elements are zero
• Symmetric: A = A^T

Matrix Operations

Addition & Subtraction

Element-wise operations requiring same dimensions.

(A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ

Multiplication

Row-column multiplication with dimension constraints.

(AB)ᵢⱼ = Σ Aᵢₖ × Bₖⱼ

Determinant

Scalar value for square matrices indicating invertibility.

det(A) ≠ 0 → invertible

Transpose

Reflection across the main diagonal.

(A^T)ᵢⱼ = Aⱼᵢ

Inverse

Matrix that when multiplied gives identity matrix.

A × A⁻¹ = I

Rank

Number of linearly independent rows or columns.

rank(A) ≤ min(m,n)

Applications

• Computer graphics transformations
• Solving systems of equations
• Data analysis and statistics
• Machine learning algorithms
• Physics and engineering
• Economics and optimization

Important Rules

• Matrix multiplication is not commutative
• (AB)^T = B^T A^T
• det(AB) = det(A) × det(B)
• (A^-1)^T = (A^T)^-1
• Rank is preserved under row operations

Common Errors

• Dimension mismatch in operations
• Assuming multiplication is commutative
• Dividing by matrices instead of multiplying by inverse
• Confusing row and column operations
• Incorrect determinant calculation

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