1. What is Matrix Dot Product?

The matrix dot product (or 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 performs standard matrix multiplication:

Where:

• \( A \) — First matrix (m×n)
• \( B \) — Second matrix (n×p)
• \( C \) — Resulting matrix (m×p)
• \( c_{ij} \) — Element at row i, column j of C

It also calculates the trace of Aᵀ·B (sum of diagonal elements of the product of A transpose and B).

3. Importance of Matrix Multiplication

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

4. Using the Calculator

Instructions:

1. Enter matrix A with elements separated by spaces and rows separated by newlines
2. Enter matrix B in the same format
3. The number of columns in A must equal the number of rows in B
4. Click Calculate to see the product matrix and trace(Aᵀ·B)

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and matrix multiplication?
A: For vectors, dot product is a scalar. For matrices, it's a matrix product (different from element-wise multiplication).

Q2: What is the trace of a matrix?
A: The trace is the sum of the diagonal elements of a square matrix. Trace(Aᵀ·B) is useful in various mathematical contexts.

Q3: Can I multiply non-square matrices?
A: Yes, as long as the number of columns in the first matrix matches the number of rows in the second matrix.

Q4: What's the computational complexity of matrix multiplication?
A: Naive implementation is O(n³) for n×n matrices. More efficient algorithms exist (e.g., Strassen's algorithm).

Q5: How are matrices represented in the input?
A: Each line represents a row, with elements separated by spaces. Example: "1 2 3" on first line and "4 5 6" on second line creates a 2×3 matrix.