1. What is Matrix Multiplication?

Matrix multiplication is a binary operation that produces a matrix from two matrices. For 3×3 matrices, the product is another 3×3 matrix where each element is computed as the dot product of the corresponding row from the first matrix and column from the second matrix.

2. How Does Matrix Multiplication Work?

The calculator uses the standard matrix multiplication formula:

Where:

• \( A \) — First 3×3 matrix
• \( B \) — Second 3×3 matrix
• \( i \) — Row index (1 to 3)
• \( j \) — Column index (1 to 3)
• \( k \) — Summation index (1 to 3)

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 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 of science and engineering.

4. Using the Calculator

Tips: Enter values for both 3×3 matrices A and B. The calculator will compute the product matrix A × B. All fields must contain valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: Is matrix multiplication commutative?
A: No, matrix multiplication is not commutative. A×B is generally not equal to B×A.

Q2: What is the identity matrix for 3×3 matrices?
A: The identity matrix I has 1s on the diagonal and 0s elsewhere. For any matrix A, A×I = I×A = A.

Q3: Can I multiply matrices of different sizes?
A: This calculator is for 3×3 matrices only. In general, you can multiply m×n and n×p matrices.

Q4: What applications use matrix multiplication?
A: Computer graphics (transformations), quantum mechanics, Markov chains, neural networks, and more.

Q5: How is matrix multiplication implemented in programming?
A: Typically using nested loops as shown in this calculator's PHP code.