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 C = AB is an m×p matrix where each element is computed as the dot product of the corresponding row of A and column of B.

2. How Does Matrix Multiplication Work?

The mathematical formula for matrix multiplication is:

Where:

• \( c_{ij} \) — Element in row i, column j of the result matrix
• \( a_{ik} \) — Element in row i, column k of matrix A
• \( b_{kj} \) — Element in row k, column j of matrix B
• \( n \) — Number of columns in A (must equal number of rows in B)

3. Requirements for Matrix Multiplication

Key Requirement: The number of columns in the first matrix must equal the number of rows in the second matrix. If matrix A is m×n, matrix B must be n×p to be multipliable.

4. Using the Calculator

Instructions:

1. Enter matrix A in the first text area
2. Enter matrix B in the second text area
3. Use spaces to separate elements within a row
4. Use semicolons to separate different rows
5. Click "Calculate" to see the result

5. Frequently Asked Questions (FAQ)

Q1: Is matrix multiplication commutative?
A: No, matrix multiplication is not commutative. AB ≠ BA in general.

Q2: What happens if the matrices have incompatible dimensions?
A: The calculator will show an error message if the number of columns in A doesn't match the number of rows in B.

Q3: Can I multiply a 2×3 matrix with a 3×4 matrix?
A: Yes, the result will be a 2×4 matrix.

Q4: What's the difference between matrix multiplication and element-wise multiplication?
A: Matrix multiplication follows the dot product rule, while element-wise multiplication multiplies corresponding elements (requires identical dimensions).

Q5: How do I enter a matrix with decimal values?
A: Simply type the decimal numbers separated by spaces for columns and semicolons for rows (e.g., "1.5 2.3;4.1 5.9").