Matrix Multiplication Calculator – Multiply Matrices Online

Multiply any two compatible matrices with our free online matrix multiplication calculator. See the full product matrix with step-by-step row-by-column computation.

Matrix A Size:

Rows:

Cols:

Current: 2x2

Matrix B Size:

Rows:

Cols:

Current: 2x2

Examples:

Matrix A (2x2)

Enter matrix values (each row on a new line)

Current: 2x2 | Detected from input

Matrix B (2x2)

Enter matrix values (each row on a new line)

Current: 2x2 | Detected from input

Understanding Matrix Multiplication

Matrix multiplication is fundamentally different from matrix addition. Instead of element-wise operations, each entry in the result is computed by taking the dot product of a row from the first matrix with a column from the second matrix.

The key rule: you can only multiply matrix A (m x n) by matrix B (n x p) when the number of columns in A equals the number of rows in B. The result is an m x p matrix.

How Matrix Multiplication Works

The Formula

If A is m x n and B is n x p, then C = AB is m x p
C[i,j] = sum of (A[i,k] x B[k,j]) for k = 1 to n

Not Commutative

AB is usually not equal to BA. Order matters!

Associative

(AB)C = A(BC). Grouping doesn't matter.

Distributive

A(B + C) = AB + AC. Works like regular algebra.

Worked Examples

Example 1: 2x2 Multiplication

A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]

C[0,0] = 1*5 + 2*7 = 5 + 14 = 19

C[0,1] = 1*6 + 2*8 = 6 + 16 = 22

C[1,0] = 3*5 + 4*7 = 15 + 28 = 43

C[1,1] = 3*6 + 4*8 = 18 + 32 = 50

AB = [[19, 22], [43, 50]]

Example 2: 2x3 times 3x2

A = [[1, 2, 3], [4, 5, 6]], B = [[7, 8], [9, 10], [11, 12]]

C[0,0] = 1*7 + 2*9 + 3*11 = 7 + 18 + 33 = 58

C[0,1] = 1*8 + 2*10 + 3*12 = 8 + 20 + 36 = 64

C[1,0] = 4*7 + 5*9 + 6*11 = 28 + 45 + 66 = 139

C[1,1] = 4*8 + 5*10 + 6*12 = 32 + 50 + 72 = 154

AB = [[58, 64], [139, 154]]

Example 3: Identity Matrix

I = [[1, 0], [0, 1]], A = [[3, 4], [5, 6]]

IA = [[1*3+0*5, 1*4+0*6], [0*3+1*5, 0*4+1*6]]

IA = [[3, 4], [5, 6]] = A

Identity matrix leaves A unchanged!

Quick Fact

Matrix multiplication seems backwards at first – rows times columns. This convention was chosen because it makes matrix multiplication correspond to function composition. When matrices represent linear transformations, AB means "apply B first, then A" – just like f(g(x)) in function notation.

Frequently Asked Questions

Why must columns of A equal rows of B?

Each result element is a dot product of a row from A with a column from B. Dot products require vectors of equal length. The row length from A must match the column length from B.

Is matrix multiplication commutative?

Generally no! AB usually does not equal BA. Sometimes one product isn't even defined when the other is. Even for square matrices where both are defined, they typically give different results.

Can you multiply a matrix by itself?

Yes, but only if it's square! A square matrix (n x n) can always be multiplied by itself because the columns equal the rows. This is called squaring the matrix: A^2 = AA.

What happens when you multiply by the identity matrix?

The identity matrix acts like the number 1 in regular multiplication. AI = A and IA = A for any compatible matrix A. The identity leaves other matrices unchanged.

What are real-world applications?

Matrix multiplication powers computer graphics (transforming 3D objects), machine learning (neural network layers), physics simulations, economics models, network analysis, and cryptography algorithms.

How do you multiply large matrices efficiently?

Standard multiplication is O(n^3) for n x n matrices. Advanced algorithms like Strassen's algorithm can do it faster for very large matrices. GPUs excel at parallel matrix multiplication.

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