1. What is Matrix Multiplication?

Matrix multiplication is a binary operation that produces a matrix from two matrices. For matrices A (m×n) and B (n×p), their product AB is an m×p matrix where each element is the sum of products of elements from the corresponding row of A and column of B.

2. How Does the Calculator Work?

The calculator uses the standard matrix multiplication formula:

Where:

• \( A \) — First matrix with dimensions m×n
• \( B \) — Second matrix with dimensions n×p
• \( (AB)_{ij} \) — Element at row i, column j of the product matrix
• The sum is taken over the index \( k \) from 1 to n

Explanation: Each element of the resulting matrix is computed by multiplying elements of the corresponding row from the first matrix with elements of the corresponding column from the second matrix and summing the products.

3. Importance of Matrix Multiplication

Details: Matrix multiplication is fundamental in linear algebra and has applications in computer graphics, physics simulations, machine learning, and many other fields. It's particularly useful for representing and solving systems of linear equations.

4. Using the Calculator

Tips: Enter matrices using semicolons to separate rows and commas to separate elements within a row. For example, "a,b;c,d" represents a 2×2 matrix. The number of columns in the first matrix must match the number of rows in the second matrix.

5. Frequently Asked Questions (FAQ)

Q1: Can I multiply any two matrices?
A: No, the number of columns in the first matrix must equal the number of rows in the second matrix.

Q2: Does matrix multiplication commute?
A: Generally no, AB ≠ BA. Matrix multiplication is not commutative.

Q3: What's the difference between element-wise and matrix multiplication?
A: Element-wise multiplication multiplies corresponding elements, while matrix multiplication follows the sum-of-products rule.

Q4: Can I use numbers and symbols together?
A: Yes, the calculator supports both numeric and symbolic entries (like 'x', 'y', etc.).

Q5: What's the time complexity of matrix multiplication?
A: For n×n matrices, the standard algorithm is O(n³), though more efficient algorithms exist.