📊 Matrix Multiplication Calculator

Matrix multiplication is a binary operation that takes a pair of matrices and produces another matrix. Unlike simple arithmetic multiplication, matrix multiplication follows specific rules and has requirements on the dimensions of the matrices.

⚠️ Two matrices can be multiplied only when the number of columns in the first matrix equals the number of rows in the second matrix.

To calculate the element at position (i, j) in the resulting matrix:

1. Take the i^th row of the first matrix
2. Take the j^th column of the second matrix
3. Multiply corresponding elements and sum them up

Matrix multiplication is used across numerous fields:

1. Enter the number of rows and columns for Matrix A
2. Click "Update" to generate the input grid
3. Enter the number of rows and columns for Matrix B
4. Click "Update" to generate the input grid
5. Fill in the values for each matrix cell

This matrix multiplier calculator follows the standard rules for multiplying matrices. If you need to calculate a matrix product for matrices of different sizes, make sure they are compatible for multiplication.

You can input:

• Integers: 1, 2, 3, etc.
• Decimals: 1.5, 2.75, etc.
• Fractions: 1/2, 3/4, etc.

Need random values? Click the "Random Values" button to populate both matrices with random numbers between 1 and 10.

After entering your matrices:

1. Click the "Calculate A × B" button
2. The result matrix will appear in the result section
3. Review the step-by-step explanation to understand how each element was calculated

Matrix multiplication requires that the number of columns in the first matrix equals the number of rows in the second matrix. This is because each element in the result is calculated by taking the dot product of a row from the first matrix and a column from the second matrix, which requires them to be the same length.

Matrix multiplication follows specific rules as explained in this calculator, resulting in a matrix where each element is the sum of products of corresponding row and column elements.

Element-wise multiplication (sometimes called the Hadamard product) simply multiplies corresponding elements directly. For element-wise multiplication, both matrices must be the same size, and the result will also be that size.

Unlike regular number multiplication where a×b = b×a, matrix multiplication is not commutative because the operation depends on the specific arrangement of elements in the matrices. The dimensions and internal structure of matrices make A×B and B×A different operations that generally yield different results.

In some special cases (like when multiplying by an identity matrix), matrix multiplication can be commutative, but this is the exception rather than the rule.

The standard algorithm for multiplying an m×n matrix by an n×p matrix has a time complexity of O(m×n×p). For square matrices of size n×n, this becomes O(n³).

However, there are more efficient algorithms for large matrices:

• Strassen's algorithm: O(n^2.807)
• Coppersmith–Winograd algorithm: O(n^2.376)

For practical purposes with small to medium-sized matrices (like those in this calculator), the standard algorithm is usually sufficient.

While this calculator specifically focuses on matrix multiplication, solving systems of linear equations typically involves multiple matrix operations including:

1. Representing the system as a matrix equation Ax = b
2. Finding the inverse of A (if it exists)
3. Computing x = A⁻¹b

For a full solution to systems of linear equations, you would need additional operations like matrix inversion or Gaussian elimination.