What can this linear algebra calculator help with?

Enter matrix operations, linear systems, row reduction, rank, null spaces, and vector questions. It generates AI explanations rather than running a dedicated exact-arithmetic matrix engine.

How do I enter a matrix?

Use nested brackets, for example [[2,1],[1,3]]. Include every row, preserve minus signs, and name the operation. You can also upload a readable image.

Can I ask for RREF or Gaussian elimination?

Yes. State the method, include the augmented column when solving a system, and ask the solver to identify every row operation and free variable.

Is the calculator free, and does it show steps?

You can start without signing up, subject to the site’s usage limits. Ask for a worked explanation and verify the AI-generated result independently.

Matrices, methods, and checks

Linear Algebra Calculator

Use this linear algebra calculator for AI explanations of matrix operations, row reduction, Ax = b, rank, and vectors. Paste a matrix or upload a problem and review the steps.

Type or upload a matrixWorked explanationsNo sign-up required

Enter a problem

linear algebra calculator

Ax = b

Matrix operations

Systems and row reduction

Rank and vector spaces

linear algebra calculator

Work through your linear algebra problem

Paste each matrix row in brackets, or upload a clear image. State the operation and ask for steps. Responses are AI-generated; check important arithmetic independently.

Text inputImage upload

Drop files to Attach, orbrowse

linear algebra calculator

Choose the linear algebra operation you need

Matrix arithmetic, systems, and vectors need different checks. Specify the task as well as the entries.

01A·B / A⁻¹

Matrix operations

Ask for addition, multiplication, a transpose, or an inverse. For AB, the columns of A must match the rows of B.

02[A | b] → RREF

Systems and row reduction

Solve Ax = b by Gaussian elimination or reduce an augmented matrix to RREF. Identify unique, inconsistent, and underdetermined systems.

03rank(A) + nullity(A) = n

Rank and vector spaces

Use pivots to find rank and free variables to describe a null space. Ask for a basis and a check of linear independence.

04u·v / Av = λv

Vectors and spectral questions

Ask about dot products, projections, determinants, or eigenvectors. The focused calculators below cover determinant and eigenvalue methods.

Method

How to use the linear algebra calculator

Keep the original matrix visible while you follow the explanation.

1
Enter the matrices
A, b
Use A = [[2,1],[1,3]] and b = [5,5]. Each inner bracket is one row.
2
Name the task
Ax = b
Specify AB, an inverse, RREF, a basis, or solving Ax = b.
3
Follow the method
R₂ ← R₂ − ½R₁
Track row operations and distinguish pivot variables from free variables.
4
Verify the output
Ax − b = 0
Substitute x into Ax = b, multiply A by its inverse, or test each basis vector.

Tip: keep fractions such as 1/3 instead of rounded decimals, and state whether parameters are real or complex.

Quick reference

Matrix dimensions and verification

Choose a check that matches the operation, rather than treating every answer as a single number.

Matrix product

(m×n)(n×p) → m×p

Multiply each row of A by each column of B; AB generally differs from BA.

Inverse

AA⁻¹ = A⁻¹A = I

Only a nonsingular square matrix has a two-sided inverse.

Linear system

Ax = b

A zero coefficient row with a nonzero augmented entry means no solution.

Rank–nullity

rank(A) + nullity(A) = n

For a matrix with n columns, count pivots and free variables.

Worked example

Worked linear algebra examples

These reviewed examples show the method, result, and an independent check. Use the example prompts above to try them in the solver.

Worked example 1

Solve Ax = b: A = [[2,1],[1,3]], b = [5,5]

Row reduction connects a matrix to its original equations.

Write 2x + y = 5 and x + 3y = 5.
Apply R₂ ← R₂ − (1/2)R₁ to [A | b]: the second row becomes [0, 5/2 | 5/2].
The second equation gives y = 1; substitute into 2x + y = 5 to get x = 2.

Result: x = [2,1]

Check: A[2,1] = [2·2 + 1, 2 + 3·1] = [5,5] = b.

Worked example 2

Find A⁻¹: A = [[2,1],[1,3]]

The same matrix lets you connect the inverse to solving a system.

det(A) = 2·3 − 1·1 = 5, so an inverse exists.
For a 2×2 matrix, swap the diagonal entries and negate the off-diagonal entries.
Divide every entry by 5: A⁻¹ = (1/5)[[3,−1],[−1,2]].

Result: A⁻¹ = [[3/5,−1/5],[−1/5,2/5]]

Check: AA⁻¹ = (1/5)[[5,0],[0,5]] = I; also A⁻¹[5,5] = [2,1].

Worked example 3

Find rank and null space: A = [[1,2,3],[2,4,6]]

A dependent row leaves free variables.

Apply R₂ ← R₂ − 2R₁ to get [[1,2,3],[0,0,0]].
One pivot gives rank(A) = 1. Write x₁ = −2x₂ − 3x₃.
Set x₂ = s and x₃ = t: x = s[−2,1,0] + t[−3,0,1].

Result: rank(A) = 1; null-space basis = {[−2,1,0], [−3,0,1]}

Check: A[−2,1,0] = 0 and A[−3,0,1] = 0; rank 1 + nullity 2 = 3 columns.

Common mistakes to catch

Check the conditions before trusting a result.

Multiplying incompatible shapes

AB exists only when the inner dimensions match. Matrix multiplication also depends on order.

Taking an inverse of any matrix

A rectangular matrix has no two-sided inverse; a square matrix with determinant zero is singular.

Confusing free variables with failure

Free variables can describe infinitely many solutions. Check for inconsistency separately.