Linear Algebra
Matrix Calculator
Add, subtract, and multiply matrices — or transpose, invert, raise to a power, and find the determinant and trace of a square one. Every result updates live, with the working shown.
Build your matrices
A × BTwo matrices
One matrix
A rows (m)
2
Inner (n)
2
B cols (p)
2
Matrix A (2 × 2)
Matrix B (2 × 2)
Result
A × B =
19
22
43
50
2×2 matrix
Updates as you type.
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The basics
What is a matrix?
A matrix is a rectangular grid of numbers arranged in rows and columns. Its shape is written rows × columns — a matrix with 2 rows and 3 columns is a 2×3 matrix. Matrices package up systems of equations, transformations, and data, and the operations below are how you compute with them.
A(i, j)
Entries and shape
Each number is an entry addressed by its row i and column j. The shape decides which operations are even allowed.
A 2×3 matrix has 6 entries
Square: rows = columns
Square matrices are special
Determinant, inverse, trace, and powers only make sense when a matrix is square — the same number of rows as columns.
2×2, 3×3, 4×4 …
Method
How matrix multiplication works
Multiplication is the operation people most often get wrong, so here it is in three steps.
- 1
Check the inner dimensions
Line up A (m×n) next to B (n×p). The two inner numbers must match, and the result will be m×p.
A is 2×3, B is 3×2 → C is 2×2 - 2
Dot each row with each column
Take row i of A and column j of B, multiply the matching pairs, and add them. That sum is the single entry C(i, j).
C(1,1) = a₁₁·b₁₁ + a₁₂·b₂₁ + a₁₃·b₃₁ - 3
Fill every entry
Repeat for every row–column pairing until the whole product matrix is built.
C(i, j) = Σₖ A(i, k) · B(k, j)
Worked examples
See each operation worked out
The same matrices, run through the four operations students reach for most — every number below is computed, not typed in.
Given A = [ 1 2 | 3 4 ] B = [ 5 6 | 7 8 ]
Add matching cells: C(1,1) = 1 + 5 = 6
Repeat across all 2×2 positions.
A + B = [ 6 8 | 10 12 ]
Reference
Every operation at a glance
What each operation needs and the shape it returns. The tool above covers all of them.
| Operation | Symbol | Requires | Result |
|---|---|---|---|
| Addition | A + B | Same shape (m×n and m×n) | m × n |
| Subtraction | A − B | Same shape (m×n and m×n) | m × n |
| Scalar multiply | k · A | Any matrix | m × n |
| Multiplication | A × B | Inner dims match (m×n · n×p) | m × p |
| Transpose | Aᵀ | Any matrix | n × m |
| Power | Aⁿ | Square matrix | n × n |
| Determinant | det A | Square matrix | scalar |
| Inverse | A⁻¹ | Square, non-singular (det ≠ 0) | n × n |
| Trace | tr A | Square matrix | scalar |
Quick chart
Can you multiply these shapes?
Pick the shape of A down the side and B across the top. A shaded cell means the product is defined; it shows the shape of the result. Blank means the inner dimensions don't match, so A × B is undefined.
| A ＼ B | 1×1 | 1×2 | 1×3 | 1×4 | 2×1 | 2×2 | 2×3 | 2×4 | 3×1 | 3×2 | 3×3 | 3×4 | 4×1 | 4×2 | 4×3 | 4×4 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1×1 | 1×1 | 1×2 | 1×3 | 1×4 | — | — | — | — | — | — | — | — | — | — | — | — |
| 1×2 | — | — | — | — | 1×1 | 1×2 | 1×3 | 1×4 | — | — | — | — | — | — | — | — |
| 1×3 | — | — | — | — | — | — | — | — | 1×1 | 1×2 | 1×3 | 1×4 | — | — | — | — |
| 1×4 | — | — | — | — | — | — | — | — | — | — | — | — | 1×1 | 1×2 | 1×3 | 1×4 |
| 2×1 | 2×1 | 2×2 | 2×3 | 2×4 | — | — | — | — | — | — | — | — | — | — | — | — |
| 2×2 | — | — | — | — | 2×1 | 2×2 | 2×3 | 2×4 | — | — | — | — | — | — | — | — |
| 2×3 | — | — | — | — | — | — | — | — | 2×1 | 2×2 | 2×3 | 2×4 | — | — | — | — |
| 2×4 | — | — | — | — | — | — | — | — | — | — | — | — | 2×1 | 2×2 | 2×3 | 2×4 |
| 3×1 | 3×1 | 3×2 | 3×3 | 3×4 | — | — | — | — | — | — | — | — | — | — | — | — |
| 3×2 | — | — | — | — | 3×1 | 3×2 | 3×3 | 3×4 | — | — | — | — | — | — | — | — |
| 3×3 | — | — | — | — | — | — | — | — | 3×1 | 3×2 | 3×3 | 3×4 | — | — | — | — |
| 3×4 | — | — | — | — | — | — | — | — | — | — | — | — | 3×1 | 3×2 | 3×3 | 3×4 |
| 4×1 | 4×1 | 4×2 | 4×3 | 4×4 | — | — | — | — | — | — | — | — | — | — | — | — |
| 4×2 | — | — | — | — | 4×1 | 4×2 | 4×3 | 4×4 | — | — | — | — | — | — | — | — |
| 4×3 | — | — | — | — | — | — | — | — | 4×1 | 4×2 | 4×3 | 4×4 | — | — | — | — |
| 4×4 | — | — | — | — | — | — | — | — | — | — | — | — | 4×1 | 4×2 | 4×3 | 4×4 |
Where it shows up
Why matrices matter
Behind a grid of numbers sits most of modern computation and science.
v' = M v
Graphics & games
Rotations, scaling, and 3D-to-2D projection are all matrix multiplications applied to coordinate vectors.
Ax = b
Solving systems
A whole system of linear equations collapses to a single matrix equation; the inverse or row reduction solves it.
W·x + b
Machine learning
Every layer of a neural network is a matrix multiply followed by a bias — the core operation of deep learning.
Properties
Rules worth remembering
A handful of identities — and one famous non-identity — make matrix algebra predictable.
AB ≠ BA
Not commutative
Order matters in multiplication. Swapping the factors usually changes the answer and can make it undefined.
(AB)C = A(BC)
Associative
You can regroup a chain of products however you like — the result is the same.
A(B + C) = AB + AC
Distributive
Multiplication distributes over addition on both the left and the right.
(Aᵀ)ᵀ = A
Transpose twice
Transposing a matrix and then transposing again returns the original.
(AB)ᵀ = BᵀAᵀ
Transpose of a product
The transpose of a product reverses the order of the factors.
A · A⁻¹ = I
Inverse undoes the matrix
A square, non-singular matrix times its inverse gives the identity — the matrix version of 1.
Watch out
Common mistakes
The errors that cost the most marks on a linear algebra exam.
Multiplying entry-by-entry
Multiplication is not elementwise. Only addition and subtraction combine matching cells; multiplication is row-times-column.
Assuming AB = BA
Reversing the order is one of the most common slips. Compute the product in the order the problem gives you.
Inverting a non-square matrix
Inverse, determinant, and trace only apply to square matrices. Check the shape before you start.
Inverting a singular matrix
If the determinant is zero, there is no inverse — no method will produce one. Check det ≠ 0 first.
FAQ
Common questions
Matrix addition and subtraction are elementwise: combine the entries that sit in matching positions, so C(i, j) = A(i, j) ± B(i, j). Both matrices must have exactly the same dimensions — a 2×3 can only be added to another 2×3. If the shapes differ, the operation is undefined.
Each entry of the product is the dot product of a row of the first matrix with a column of the second: multiply the matching pairs and add them. So C(1, 1) = a₁₁·b₁₁ + a₁₂·b₂₁ + …. The product A × B is only defined when the number of columns of A equals the number of rows of B.
Each output entry pairs a row of A with a column of B one number at a time. A row of A has as many numbers as A has columns, and a column of B has as many numbers as B has rows. To pair them up exactly, those two counts — the inner dimensions — must be equal. If A is m×n and B is n×p, the product is m×p.
No. In general AB ≠ BA — order matters, and swapping the factors can change the answer or even make the product undefined. Matrix multiplication is associative ( (AB)C = A(BC) ) and distributive over addition, but never assume it commutes.
The determinant is a single number computed from a square matrix that tells you whether the matrix is invertible. A determinant of zero means the matrix is singular and has no inverse. For a 2×2 matrix [[a, b], [c, d]] the determinant is ad − bc; larger matrices use cofactor (Laplace) expansion.
Only square matrices with a non-zero determinant have an inverse. The standard method augments the matrix with the identity, [A | I], then applies Gauss–Jordan row reduction until the left block becomes the identity — at which point the right block is A⁻¹. The inverse satisfies A · A⁻¹ = I.
The transpose Aᵀ flips a matrix over its main diagonal: rows become columns and columns become rows, so Aᵀ(i, j) = A(j, i). An m×n matrix becomes n×m. Transposing twice returns the original matrix.
The trace is the sum of the entries on the main diagonal of a square matrix: tr(A) = a₁₁ + a₂₂ + … + aₙₙ. It equals the sum of the eigenvalues and is invariant under similarity transformations, which makes it useful in linear algebra and physics.
Keep going
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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