Matrix Calculator
The Matrix Calculator is a powerful linear algebra tool that allows you to perform complex matrix operations instantly. Set your dimensions, input your elements, and compute results without manual errors.
This tool supports matrices of custom sizes (up to 5x5) and handles both single-matrix transformations and multi-matrix arithmetic including:
- Determinants (|A|) and Inverses (A⁻¹)
- Transposition (Aᵀ)
- Matrix Addition, Subtraction, and Multiplication
Why use this calculator?
Calculating the inverse of a 3x3 or 4x4 matrix by hand is tedious and prone to sign errors. This tool automates the Gaussian Elimination or Adjoint Method to give you precise results in seconds.
Single Matrix Operations
These operations apply to one matrix at a time (Matrix A or Matrix B).
Determinant (|A|)
A scalar value that describes the properties of the matrix. It is critical for finding the inverse and solving systems of linear equations (Cramer's Rule).
- If |A| = 0, the matrix is "Singular" (no inverse exists).
- Requirement: Matrix must be Square (NxN).
Inverse (A⁻¹)
The matrix equivalent of division. Multiplying a matrix by its inverse results in the Identity Matrix (I).
- Used heavily in decoding encryption and 3D graphics.
- Requirement: Matrix must be Square and Non-Singular (|A| ≠ 0).
| Operation | Symbol | Description |
|---|---|---|
| Transpose | Aᵀ | Flips the matrix over its diagonal. Rows become columns, and columns become rows. |
| Rank | rk(A) | The maximum number of linearly independent row or column vectors. |
Matrix Arithmetic (A & B)
Perform operations between two matrices. Note that dimension compatibility is strict in linear algebra.
Addition & Subtraction (A ± B)
Adds or subtracts corresponding elements from two matrices.
Rule: Both matrices must have the exact same dimensions (e.g., 3x3 and 3x3).
Matrix Multiplication (A × B)
Calculates the dot product of rows and columns. This is not element-wise multiplication.
⚠️ Multiplication Rule
You can only multiply Matrix A and Matrix B if:
Columns of A = Rows of B
Example: You can multiply a (2x3) matrix by a (3x2) matrix.
You cannot multiply a (2x3) by a (2x3).
How to Use the Matrix Solver
Follow these simple steps to solve linear algebra problems instantly.
- Set Dimensions: Use the input boxes (e.g., 3x3) to define the size of Matrix A and Matrix B. Click "Set Matrix Size" to generate the grid.
- Enter Values: Input your numbers into the cells. You can use:
- Integers (e.g.,
5,-3) - Decimals (e.g.,
1.5,0.04)
- Integers (e.g.,
- Choose Operation:
- For single-matrix results (Inverse/Determinant), click the buttons above the specific matrix.
- For arithmetic (A × B), click the operation buttons between the matrices.
Pro Tip: Quick Fill Tools
Don't want to type every zero? Use the helper buttons below the grid:
- All 0: Resets the matrix to a Zero Matrix.
- All 1: Fills every cell with the number 1.
- Rand: Fills the matrix with random integers (great for testing or practice).
Where are Matrices Used in Real Life?
Matrices are the backbone of modern computing and engineering. They aren't just numbers in brackets; they are instructions for transforming data.
🎮 3D Graphics & Gaming
Every time a character moves, rotates, or scales in a video game, a Transformation Matrix is applied to their coordinates. GPU hardware is essentially a giant matrix calculator.
⚡ Electrical Circuits
Engineers use matrices to solve complex circuits (Mesh Analysis).
A system of linear equations like Ax = B helps calculate currents and voltages in a grid.
🔐 Cryptography
Matrices are used in encryption algorithms (like the Hill Cipher). You encrypt a message by multiplying it by a key matrix, and decrypt it by multiplying by the Inverse of that key.
Frequently Asked Questions
A "Singular Matrix" is a matrix that has a Determinant of 0. Mathematically, it has no inverse (it's like trying to divide by zero). You usually encounter this if one row is a perfect multiple of another row.
Matrix multiplication follows the "Inner Dimensions" rule.
The number of Columns in Matrix A must equal the number of Rows in Matrix B.
Example: (2×3) multiplied by (3×2) works. (2×3) multiplied by (2×3) fails.
No. Determinants (and Inverses) are only defined for Square Matrices (where rows equal columns, e.g., 2x2, 3x3, 4x4).
This is due to "floating-point arithmetic" in computers.
While our calculator rounds results for display, complex division operations (like finding an inverse) sometimes result in tiny rounding artifacts.
Usually, 3.000000004 can be treated as 3.