How do you multiply two matrices?

To multiply matrix A (m x n) by matrix B (n x p): 1. Check dimensions: columns of A must equal rows of B 2. The result C has dimensions m x p 3. Each cell C[i,j] = sum of A[i,k] * B[k,j] for k = 1 to n Example: A = [[1,2],[3,4]], B = [[5,6],[7,8]] C[1,1] = 1*5 + 2*7 = 5 + 14 = 19 C[1,2] = 1*6 + 2*8 = 6 + 16 = 22 C[2,1] = 3*5 + 4*7 = 15 + 28 = 43 C[2,2] = 3*6 + 4*8 = 18 + 32 = 50 Result: [[19,22],[43,50]]

Why does matrix multiplication require matching inner dimensions?

Matrix multiplication A x B is defined as the dot product of each row of A with each column of B. For a dot product to exist between a row and a column, they must have the same length. If A is m x n, each row of A has n elements. If B is n x p, each column of B has n elements. They match, so the dot product is defined. If A is 2x3 and B is 2x3, columns of A = 3 but rows of B = 2. They do not match, so multiplication is undefined. The result dimension is always (rows of A) x (columns of B).

Is matrix multiplication commutative?

No. In general, A x B is not equal to B x A. Counter-example: A = [[1,2],[3,4]], B = [[0,1],[1,0]] A x B = [[2,1],[4,3]] B x A = [[3,4],[1,2]] These are different. Additionally, A x B might be defined while B x A is not. If A is 2x3 and B is 3x4: - A x B is valid (2x4 result) - B x A requires B(3x4) x A(2x3): inner dimensions are 4 and 2, which do not match Matrix multiplication is associative (A x B) x C = A x (B x C) and distributive A x (B+C) = AB + AC, but not commutative.

What is an identity matrix and what happens when you multiply by it?

The identity matrix I is a square matrix with 1s on the diagonal and 0s everywhere else: 2x2: [[1,0],[0,1]] 3x3: [[1,0,0],[0,1,0],[0,0,1]] Multiplying any matrix A by the appropriate identity matrix leaves A unchanged: A x I = I x A = A This is analogous to multiplying a number by 1. The identity matrix is the matrix equivalent of the number 1. Try the Identity 3x3 example in this calculator to verify: multiplying any 3x3 matrix by I3 returns the original matrix exactly.

What are real-world applications of matrix multiplication?

Matrix multiplication is fundamental throughout science, engineering, and computing: Computer graphics: 3D transformations (rotation, scaling, translation) are applied by multiplying a transformation matrix by a vertex coordinate vector. Every frame rendered in a 3D game uses matrix multiplication. Machine learning: neural network layers compute y = W x + b where W is a weight matrix. Forward and backward passes are dominated by matrix multiplications. Physics simulations: systems of linear equations (Ax = b) are solved using matrix operations. Quantum mechanics uses matrix representations of operators. Economics: input-output models (Leontief models) use matrix multiplication to model how industries depend on each other. Cryptography: some encryption schemes use matrix multiplication over finite fields. Signal processing: convolutions and Fourier transforms can be expressed as matrix multiplications.

Matrix Multiplication Calculator — Step-by-Step

The Matrix Multiplication Calculator multiplies any two matrices up to 4x4 with full step-by-step working. Enter your matrices, click Multiply, and see the complete dot product calculation for every cell in the result. Click any step to highlight which row and column produced that result. Supports 2...

MATRIX A SIZE

RESULT SIZE

2×2

A(2×2) × B(2×2)

MATRIX B SIZE

Matrix A (2×2)

Matrix B (2×2)

QUICK EXAMPLES

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HOW MATRIX MULTIPLICATION WORKS

Dimension rule

A (m×n) × B (n×p) = C (m×p). The number of columns in A must equal the number of rows in B. The result has the same number of rows as A and columns as B.

Dot product

Each cell C[i,j] is the dot product of row i of A and column j of B: multiply corresponding elements and sum them all.

Not commutative

A×B ≠ B×A in general. Order matters. Swapping the matrices usually gives a completely different result — or may not even be possible.

Time complexity

Standard multiplication of an m×n matrix by an n×p matrix takes O(m×n×p) operations. A 4×4 × 4×4 requires 4×4×4 = 64 multiplications and 48 additions.

HOW TO USE

1
Set the size of Matrix A using the row and column buttons (1 to 4). The number of columns in A automatically sets the number of rows in B — this is the dimension rule for matrix multiplication.
2
Set the number of columns in Matrix B. The result matrix C will have the same rows as A and same columns as B. The result size is shown between the two size selectors.
3
Enter values in Matrix A and Matrix B. Use the quick example buttons to load common cases: 2x2, 2x3 x 3x2, identity matrix, or dot product (1x3 x 3x1).
4
Click MULTIPLY A x B. The result matrix C appears on a black background. Each cell is calculated as the dot product of the corresponding row from A and column from B.
5
Click any step in the step-by-step panel to highlight which row of A and column of B produced that result cell. The formula shows every multiplication and addition in full.

WORKED EXAMPLE

A = [[1,2],[3,4]], B = [[5,6],[7,8]]. C[1,1] = 1*5 + 2*7 = 5+14 = 19. C[1,2] = 1*6 + 2*8 = 6+16 = 22. C[2,1] = 3*5 + 4*7 = 15+28 = 43. C[2,2] = 3*6 + 4*8 = 18+32 = 50. Result C = [[19,22],[43,50]]. Dimensions: A(2x2) x B(2x2) = C(2x2).

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.