Matrix Calculator
This Matrix Calculator helps you add, subtract, multiply, transpose, and analyze matrices (determinant, inverse, RREF), plus solve linear systems Ax = b for real-world linear algebra. It’s handy for engineering, data science, and graphics transforms—without leaving your browser. For deeper inverse checks, try the inverse matrix calculator or the Algebra Calculator. You can also explore the Math Calculators hub or browse All Calculators.
Accuracy & Method
Calculations run locally in your browser using pivoted elimination (tolerance-based). No external requests.
Privacy
Inputs never leave your device. Copy features only use your clipboard with your action.
Rounding
Outputs round to your selected precision (0–6). Near-zero values are cleaned to 0 via tolerance.
Matrix Calculator Tool
Matrix A
Enter numbers (integers or decimals). Use arrow keys to move.
Matrix B
Shown only for operations that need B (A±B, A×B).
Scalar
Used for Scalar × A.
Vector b
Used for Ax = b when A is square (n×n). Length must be n.
Tip: Start with small matrices, then scale up to 6×6.
Results
Operation Summary
Operation—
Dimensions—
Pivoting—
Tolerance1e-10
Precision3
Notes—
Steps0
Main Output
Run a calculation to see a matrix or scalar result here.
Step Highlights
Choose an operation and click Calculate. You’ll see a concise breakdown here (dot-product, elimination pivots, or Gauss-Jordan outline).
How It Works
Core Definitions
- Rows/Cols: A is m×n, B is p×q.
- Indices: i = row, j = column, k = inner sum index.
- Pivot: the selected leading entry used during elimination (with partial pivoting for stability).
- Tolerance: values with |x| < 1e-10 are treated as 0 to reduce floating noise.
Formulas Used
Matrix multiplication uses: (A×B)[i,j] = Σ(A[i,k]·B[k,j]). Determinant and inverse rely on elimination with pivoting rather than slow Laplace expansion.
Determinant via elimination: after converting A to upper triangular U, det(A) = (−1)^{swaps} · Π diag(U). Inverse uses Gauss-Jordan on the augmented matrix [A | I] until the left becomes I.
Substituted-Value Snapshot
After you calculate, this line will show a concrete substitution (for example, the dot-product for the first result cell in A×B or pivot products for det(A)).
Use Cases
- Combine linear transforms in graphics by multiplying transformation matrices.
- Check invertibility for systems modeling (nonsingular matrices have an inverse).
- Solve engineering constraint systems with Ax=b to find unknowns.
- Compute RREF to identify pivot columns, rank, and dependent equations.
- Validate linear algebra homework by verifying A·A⁻¹ ≈ I under a set precision.
Worked Examples
Example 1: Addition (A + B)
Add entry-by-entry (same dimensions required).
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]
Example 2: Multiplication (A × B)
Inner dimensions must match (A cols = B rows).
A = [[1, 2, 0], [−1, 3, 1]] (2×3)
B = [[2, 1], [0, 4], [5, −2]] (3×2)
(A×B)[1,1] = 1·2 + 2·0 + 0·5 = 2
(A×B)[1,2] = 1·1 + 2·4 + 0·(−2) = 9
A×B = [[2, 9], [3, 9]]
Example 3: Solve Ax = b
For square A, elimination finds x if a unique solution exists.
A = [[2, 1], [5, 3]]
b = [1, 2]
System: 2x₁ + x₂ = 1 and 5x₁ + 3x₂ = 2
Eliminate: multiply first by 3 → 6x₁ + 3x₂ = 3; multiply second by 1 → 5x₁ + 3x₂ = 2
Subtract → x₁ = 1; back-substitute → x₂ = −1
x = [1, −1]
Common Mistakes
- Trying A + B when A and B don’t have the same shape.
- Swapping multiplication order: A×B may be valid while B×A is not (or gives a different result).
- Assuming an inverse exists: singular matrices (determinant ≈ 0) cannot be inverted.
- Entering commas or extra symbols inside cells (use plain numbers like -2.5).
- Ignoring rounding: near-zero values can appear due to floating arithmetic—precision matters.
Quick Tips
- Check dimensions first: for A×B, A cols must equal B rows.
- Use the identity matrix as a sanity check: A×I = A.
- If inversion fails, try RREF to understand pivot structure and rank.
- Increase precision when verifying identities like A·A⁻¹ ≈ I.
- For decimals, keep the blank-as-zero toggle off unless you truly want missing entries treated as 0.
Frequently Asked Questions
1) What can this Matrix Calculator do?
It performs everyday matrix operations with real in-browser math: addition, subtraction, multiplication, scalar multiplication, transpose, determinant, inverse, RREF, and solving Ax=b. You can set sizes from 1 to 6 for rows and columns and adjust output precision. For concept cross-checks, pairing results with the Algebra Calculator is useful, especially when verifying a system solution by substitution. If you’re exploring many tools, the All Calculators page is a handy hub.
2) Why can’t I add matrices of different sizes?
Matrix addition and subtraction are entry-by-entry operations. That means each element in A must have a corresponding element in B at the same row and column position. If A is 3×2 and B is 2×3, there is no consistent way to pair entries, so A±B is undefined. This tool checks the shape before calculating and will show a clear message if the dimensions don’t match. If your goal is to combine information across shapes, you may want multiplication instead, provided the inner dimensions align.
3) Why isn’t matrix multiplication commutative?
In general, A×B and B×A represent different compositions of linear transformations. Even when both products are defined, their dot-product structure uses different row/column pairings, so the results typically differ. For example, a rotation followed by a scaling is not the same as scaling followed by rotation. This calculator enforces the rule “A cols must equal B rows” and will display shape badges so you can see when only one order is valid. The Step Highlights also show a dot-product substitution to make the difference tangible.
4) When does an inverse exist?
An inverse exists only for square matrices (n×n) that are nonsingular. Practically, that means det(A) ≠ 0, or equivalently that elimination finds a pivot in every column. If the determinant is 0 (or extremely close to 0 under tolerance), the matrix is singular and cannot be inverted because it collapses space in at least one direction. This tool uses Gauss-Jordan elimination with pivoting on [A|I]. If you want a dedicated inverse-focused workflow, see the inverse matrix calculator.
5) What does the determinant tell me?
The determinant summarizes key geometry and solvability information for square matrices. A nonzero determinant indicates the transformation is invertible and the linear system Ax=b has a unique solution for any b. The magnitude relates to area/volume scaling, while the sign indicates orientation flips. This calculator computes det(A) via pivoted elimination and reports row swaps, because each swap flips the sign. If the determinant is very close to zero, results can be sensitive to rounding, so increasing precision can help interpret near-singular cases.
6) How does solving Ax = b work here?
When you choose “Solve Linear System,” the tool expects a square A and a vector b whose length equals the number of rows. It then applies elimination with partial pivoting to the augmented system [A|b]. If a pivot cannot be found in a column, the system may have infinitely many solutions (dependent equations). If a row becomes all zeros in A but the right side is nonzero, it’s inconsistent and has no solution. For a unique solution, the calculator returns x as a vector and summarizes the pivot behavior used for numerical stability.
7) What is RREF and when should I use it?
RREF (row-reduced echelon form) is a canonical version of a matrix produced by Gauss-Jordan elimination. In RREF, each pivot is 1, pivot columns contain zeros elsewhere, and pivot positions move right as you go down rows. RREF is great for analyzing rank, identifying pivot columns, solving linear systems (including multiple solutions), and checking linear independence. This calculator computes RREF with pivoting and a small tolerance so values like 1e-12 are treated as 0. The Step Highlights summarize row operations and pivot steps without overwhelming detail.
8) What sizes are supported, and how does precision affect results?
You can use matrices from 1×1 up to 6×6, which covers most homework, engineering checks, and quick modeling tasks. Larger sizes increase computation and can amplify rounding effects, especially for determinants and inverses. Precision controls how many decimals are displayed (0–6), while a tolerance treats very small values as 0 to reduce floating noise. If you see tiny leftover values like -0.000, increase precision to inspect them—or rely on tolerance to interpret them as numerical artifacts. For structured practice, browse Math Calculators.
Sources & References
- Standard linear algebra textbooks (matrix operations, elimination, determinants, inverses).
- University lecture notes on Gauss-Jordan elimination and numerical stability (partial pivoting, tolerance).
- Common computational practice for floating-point rounding and near-zero cleanup in matrix routines.