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 dot product of a row from A and a column from B.

2. How Does the Calculator Work?

The calculator implements the standard matrix multiplication algorithm:

Where:

• \( A \) — First matrix (m×n)
• \( B \) — Second matrix (n×p)
• \( AB \) — Result matrix (m×p)
• \( i \) — Row index (1 to m)
• \( j \) — Column index (1 to p)
• \( k \) — Summation index (1 to n)

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

3. Importance of Matrix Multiplication

Details: Matrix multiplication is fundamental in linear algebra and has applications in computer graphics, scientific computing, machine learning, and many other fields. It's used for linear transformations, solving systems of equations, and representing complex operations.

4. Using the Calculator

Tips:

• Enter matrices with rows separated by newlines and columns by spaces
• Matrix A must have the same number of columns as Matrix B has rows
• Example format for a 2×3 matrix: "1 2 3\n4 5 6"
• The calculator supports integer and decimal numbers

5. Frequently Asked Questions (FAQ)

Q1: What are the requirements for matrix multiplication?
A: The number of columns in the first matrix must equal the number of rows in the second matrix.

Q2: Is matrix multiplication commutative?
A: No, in general AB ≠ BA. The order of multiplication matters.

Q3: What's the time complexity of matrix multiplication?
A: The standard algorithm is O(n³) for n×n matrices. More efficient algorithms exist but are more complex.

Q4: Can I multiply non-numeric matrices?
A: This calculator only supports numeric matrices. Abstract matrix multiplication requires different implementations.

Q5: How are errors handled?
A: The calculator checks for valid numeric input and compatible dimensions, showing clear error messages when issues are detected.