Matrix Calculator

Perform matrix operations with dimension checks, exact entries, and each row or column calculation shown.

Drag & drop or upload an image or PDF

Enter an operation, such as multiply [[1,2],[3,4]] by [[2,0],[1,2]]

Enter rows in one line, such as [[1, 2], [3, 4]]. State the operation when the page accepts more than one.

Try an example

Matrix operations begin with dimensions

A matrix is a rectangular array, and its shape controls which operations are defined. Addition and subtraction require matching dimensions. Multiplication uses row-by-column products and requires the inner dimensions to agree.

State the operation and enter each row inside brackets. The calculator can add, subtract, multiply by a scalar, multiply matrices, transpose, or row reduce. Specialized determinant, inverse, and eigenvalue pages keep those larger methods separate.

How to use the matrix calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Matrix multiplication entry rule

Each result entry is the dot product of one row from the first matrix and one column from the second.

(A B)_{ij} = k = 1 \sum n a_{ik} b_{k j}

A reliable way to work through it

Write the dimensions first

Confirm equal shapes for addition or matching inner dimensions for multiplication before doing arithmetic.

Keep the order fixed

Matrix multiplication is generally not commutative, so AB and BA are different calculations and one may be undefined.

Preserve exact entries

Use fractions through row operations and round only when the problem specifically requests a decimal result.

Worked example

Multiply A by B

A = [1023 - 1 4], B = 2 - 1 3 102

The inner dimensions match: (2 × 3)(3 × 2), so AB will be 2 × 2.

c_{11} = 1 (2) + 2 (- 1) + (- 1) (3) = - 3, c_{12} = 1 (1) + 2 (0) + (- 1) (2) = - 1

Use the first row of A with both columns of B.

c_{21} = 0 (2) + 3 (- 1) + 4 (3) = 9, c_{22} = 0 (1) + 3 (0) + 4 (2) = 8

Repeat with the second row of A.

A B = [- 3 9 - 1 8]

Place each dot product in its matching row and column.

The product is a 2 × 2 matrix with rows [-3, -1] and [9, 8].

Common mistakes to check

Multiplying matching entries

Matrix multiplication uses row-by-column dot products, not entry-by-entry multiplication.

Checking the wrong dimensions

For AB, compare the columns of A with the rows of B. The outer dimensions give the result size.

Assuming AB equals BA

Changing the order changes the row-column pairings and can even make the product undefined.

Determinant CalculatorCalculate the scalar determinant of a square matrix and track every cofactor sign or row-operation effect.Inverse Matrix CalculatorFind A⁻¹ with exact row operations, detect a singular matrix, and verify the result with the identity.Eigenvalue CalculatorFind the characteristic polynomial, eigenvalues, and eigenspaces of a square matrix.Systems of Equations CalculatorSolve two or more equations together with substitution, elimination, or row operations.

Questions students ask

Can I multiply matrices of different sizes?

Yes, when the number of columns in the first matrix equals the number of rows in the second.

Why can AB exist when BA does not?

Reversing the order changes which inner dimensions must match.

Can I use a non-square matrix?

Yes for operations such as transpose, scalar multiplication, compatible multiplication, and row reduction.

Why should I keep fractions exact?

Exact fractions prevent rounding drift from accumulating across several row operations.