Matrix Calculator | Addition, Multiplication, Transpose & More

📖 How to Use This Matrix Calculator

Set matrix size - Choose the number of rows and columns (1-4).

Enter values - Fill in the cells of Matrix A and Matrix B with numbers.

Select operation - Choose from Add, Subtract, Multiply, Transpose, Determinant, or Inverse.

View result - The calculated matrix or value appears instantly.

📊 Understanding Matrix Operations

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental in linear algebra, used for solving systems of equations, computer graphics, machine learning, and data analysis.

Matrix Addition & Subtraction

Matrices must have the same dimensions. Add/subtract corresponding elements: (A+B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ.

Matrix Multiplication

For A (m×n) and B (n×p), result is (m×p). Multiply rows of A by columns of B: (AB)ᵢⱼ = Σ Aᵢₖ × Bₖⱼ. The number of columns in A must equal rows in B.

Matrix Transpose

Swap rows and columns. If A is m×n, Aᵀ is n×m where (Aᵀ)ᵢⱼ = Aⱼᵢ.

Determinant

A scalar value that indicates whether a matrix is invertible (det ≠ 0). For 2×2: det = ad - bc. For 3×3: det = a(ei − fh) − b(di − fg) + c(dh − eg).

Inverse Matrix

A⁻¹ exists only for square matrices with non-zero determinant. A × A⁻¹ = I (identity matrix). Used for solving linear equations Ax = b → x = A⁻¹b.

💡 Applications of Matrices

Computer Graphics: Transformations (rotation, scaling, translation)
Machine Learning: Data representation and neural networks
Economics: Input-output models
Physics: Quantum mechanics, optics, and dynamics
Engineering: Circuit analysis and structural analysis

❓ Frequently Asked Questions (FAQ)

When can I multiply two matrices?

Matrix multiplication is possible when the number of columns in the first matrix equals the number of rows in the second matrix. For example, a 2×3 matrix can be multiplied by a 3×4 matrix.

What is an identity matrix?

An identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. It's the multiplicative identity: A × I = I × A = A.

What does a zero determinant mean?

A zero determinant means the matrix is singular (not invertible). The rows/columns are linearly dependent, and the matrix cannot be inverted.

Why is matrix multiplication not commutative?

In general, A×B ≠ B×A because the order of multiplication matters. Matrix multiplication represents composition of linear transformations, which is not commutative.

What is the maximum matrix size supported?

This calculator supports matrices up to 4×4. For larger matrices, dedicated scientific computing software is recommended.

How is the inverse of a 3×3 matrix calculated?

For a 3×3 matrix, we use the formula involving the adjugate matrix divided by the determinant. The calculator shows the result after computing all cofactors.