Matrix calculator
Fast, privacy-first online Matrix calculator tool. 100% in-browser client-side execution with zero signups and zero data tracking.
1. Matrices & Operations
2. Calculated Result & Expansion
Matrix Properties
Step-by-Step Mathematical Derivation
Determinant calculated via Laplace cofactor expansion along Row 1:
- +1 × |(1)(0) - (4)(6)| = 1 × (-24) = -24
- -2 × |(0)(0) - (4)(5)| = -2 × (-20) = +40
- +3 × |(0)(6) - (1)(5)| = 3 × (-5) = -15
- Total: -24 + 40 - 15 = 1.00
About this tool
How to use it
- Choose your matrix dimension: 2x2, 3x3, or 4x4 using the dimension buttons.
- Select your target operation: Determinant |A|, Inverse A⁻¹, Transpose Aᵀ, Trace, Matrix Multiplication (A × B), Addition, or Subtraction.
- Input the cell values directly into Matrix A (and Matrix B if using a dual-matrix operation).
- Inspect the real-time matrix properties strip: Determinant value, Trace (sum of diagonal elements), and Invertibility status.
- Review the step-by-step mathematical derivation showing cofactor expansion or row-by-column dot products.
- Click 'Copy Solution' to copy the calculation summary to your clipboard.
Why it matters
Linear algebra operations—especially 3x3 and 4x4 determinants and matrix inverses—require hundreds of intermediate multiplications, additions, and sign flips. A single arithmetic slip invalidates the entire matrix. An instant calculator with step-by-step expansion helps students verify homework and engineers validate transformation matrices.
When to use it
Use when studying college linear algebra, verifying engineering rotation matrices, solving systems of linear equations via Cramer's Rule, or calculating physics tensors.
Good to know
A matrix is invertible if and only if its determinant is non-zero (det ≠ 0). If det(A) = 0, the matrix is singular and possesses no multiplicative inverse.
Common questions
What is the determinant of a matrix?
The determinant is a scalar value calculated from a square matrix that characterizes the scaling factor of the transformation. If the determinant is 0, the transformation collapses dimensions and the matrix cannot be inverted.
What is the trace of a matrix?
The trace of a square matrix is the sum of the elements on the main diagonal (from top-left to bottom-right). It equals the sum of the matrix's eigenvalues.
How does matrix multiplication work?
Matrix multiplication is computed by taking the dot product of row i from the first matrix with column j from the second matrix to produce element (i, j) of the product matrix. Notice that A × B is generally not equal to B × A (non-commutative).
Can any square matrix be inverted?
No. Only square matrices with a non-zero determinant can be inverted. Matrices with det = 0 are called singular or degenerate matrices.
Does this tool work completely offline?
Yes. All linear algebra calculations execute 100% locally in your web browser memory with zero network dependencies.