Matrix Multiplication Calculator

Matrix multiplication is an operation that produces a new matrix from two input matrices. Each element of the result is the dot product of a row from the first matrix and a column from the second matrix. It is fundamentally different from element-wise multiplication.

For the product AB to be defined, the number of columns in A must equal the number of rows in B. If A is m × p and B is p × n, the result AB is an m × n matrix. If the dimensions do not match, multiplication is undefined.

No, matrix multiplication is generally NOT commutative: AB ≠ BA in most cases. The order of multiplication matters. In some special cases (like when both matrices are diagonal), commutativity may hold.

Yes, matrix multiplication is associative: (AB)C = A(BC) for any compatible matrices A, B, and C. This means you can group multiplications in any order without changing the result.

Yes, matrix multiplication distributes over addition: A(B + C) = AB + AC and (A + B)C = AC + BC. However, the order of multiplication must be preserved due to non-commutativity.

The identity matrix I acts as the multiplicative identity: AI = IA = A for any compatible matrix A. The identity matrix has 1s on the diagonal and 0s elsewhere.

The dot product is the sum of element-wise products of two vectors. In matrix multiplication, element (i,j) of the result is the dot product of row i from the first matrix and column j from the second matrix.

For A = [[a,b],[c,d]] and B = [[e,f],[g,h]], the product is AB = [[ae+bg, af+bh], [ce+dg, cf+dh]]. Each element is the dot product of the corresponding row and column.

If the number of columns in A does not equal the number of rows in B, matrix multiplication is undefined. The calculator will display an error message indicating dimension mismatch.

Yes, a vector can be treated as a matrix with one column (or one row). Multiplying an m × n matrix by an n × 1 column vector produces an m × 1 column vector.

The standard algorithm has O(m × n × p) complexity for multiplying an m × p matrix by a p × n matrix. Optimized algorithms like Strassen's reduce this to approximately O(n^2.807) for square matrices.

The Hadamard (element-wise) product multiplies corresponding elements and requires identical dimensions. Standard matrix multiplication uses row-column dot products and requires compatible (but not necessarily identical) dimensions.

Matrix multiplication corresponds to composition of linear transformations. If A transforms space and then B transforms it, the combined transformation is BA (applied right to left).

Yes, you can multiply any number of matrices sequentially, as long as adjacent matrices have compatible dimensions. Due to associativity, the grouping does not matter, but the order does.

The transpose of a product equals the product of the transposes in reverse order: (AB)ᵀ = BᵀAᵀ. This is a key property used in many matrix algebra proofs.

The inverse of a product equals the product of the inverses in reverse order: (AB)⁻¹ = B⁻¹A⁻¹, provided both A and B are invertible.

Matrix multiplication combines transformations like rotation, scaling, and translation. A single matrix can encode multiple transformations, and multiplying them applies all transformations efficiently.

Scalar multiplication multiplies every element by a single number. Matrix multiplication combines rows and columns using dot products, producing a result that depends on the structure of both matrices.

Yes, matrix multiplication is available in most linear algebra libraries. In NumPy: A @ B or np.dot(A, B). In JavaScript, you need nested loops or a library like math.js.

Matrix power Aⁿ means multiplying A by itself n times: A³ = A × A × A. For n = 0, A⁰ = I (identity matrix). For n = —1, A⁻¹ is the inverse matrix.

The Matrix Multiplication Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Matrix Multiplication Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.