Matrix Calculator Multiplication

Matrix Multiplication Formula:

\[ (A B)_{ij} = \sum_k A_{ik} B_{kj} \]

Matrix A:

Matrix B:

Result Matrix:

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1. What is Matrix Multiplication?

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

2. How Matrix Multiplication Works

The calculator uses the standard matrix multiplication formula:

\[ (A B)_{ij} = \sum_{k=1}^n A_{ik} B_{kj} \]

Where:

\( A \) — First matrix (m×n dimensions)
\( B \) — Second matrix (n×p dimensions)
\( i \) — Row index of resulting matrix
\( j \) — Column index of resulting matrix
\( k \) — Summation index

Explanation: Each element of the resulting matrix is computed as the dot product of the corresponding row from the first matrix and column from the second matrix.

3. Applications of Matrix Multiplication

Details: Matrix multiplication is fundamental in linear algebra and has applications in computer graphics, physics simulations, machine learning, and more.

4. Using the Calculator

Tips: Enter values for both 2×2 matrices. The calculator will compute the product matrix. All fields are required for accurate results.

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. Matrix multiplication is not commutative.

Q3: What's the identity matrix for multiplication?
A: The identity matrix I has 1s on the diagonal and 0s elsewhere. For any matrix A, AI = IA = A.

Q4: Can I multiply non-square matrices?
A: Yes, as long as the column count of the first matches the row count of the second.

Q5: What's the computational complexity?
A: For n×n matrices, naive multiplication is O(n³). Advanced algorithms can do better.