1. What Is Matrix Multiplication?

Matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix.

2. How Does Matrix Multiplication Work?

The matrix multiplication follows this formula:

Where:

• \( A \) — First matrix of dimensions m × n
• \( B \) — Second matrix of dimensions n × p
• \( C \) — Result matrix of dimensions m × p
• \( c_{ij} \) — Element in row i, column j of result matrix
• \( a_{ik} \) — Element in row i, column k of matrix A
• \( b_{kj} \) — Element in row k, column j of matrix B

Explanation: Each element in the resulting matrix is the dot product of a row from the first matrix and a column from the second matrix.

3. Requirements For Matrix Multiplication

Details: For matrices A (m×n) and B (p×q) to be multiplied, n must equal p. The resulting matrix will have dimensions m×q.

4. Using The Calculator

Tips: Enter matrices using comma separated values within rows and semicolon separated rows. Example: "1,2,3;4,5,6" creates a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: Is Matrix Multiplication Commutative?
A: No, matrix multiplication is not commutative. A×B ≠ B×A in most cases.

Q2: What Is The Identity Matrix?
A: An identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. When multiplied by any matrix, it returns the original matrix.

Q3: Can Any Two Matrices Be Multiplied?
A: No, only when the number of columns in the first matrix equals the number of rows in the second matrix.

Q4: What Are The Applications Of Matrix Multiplication?
A: Used in computer graphics, physics simulations, economics, machine learning, and solving systems of linear equations.

Q5: How Does Matrix Multiplication Differ From Element-wise Multiplication?
A: Matrix multiplication follows the dot product rule, while element-wise multiplication multiplies corresponding elements (requires same dimensions).