2x3 3x2 Matrix Multiplication Calculator

Matrix Multiplication:

\[ C = A \times B \text{ where } c_{ij} = \sum_{k=1}^{3} a_{ik} \times b_{kj} \]

Matrix A (2x3)

Matrix B (3x2)

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1. What Is Matrix Multiplication?

Matrix multiplication is a binary operation that produces a matrix from two matrices. For matrices to be multiplied, 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 calculator uses the standard matrix multiplication formula:

\[ c_{ij} = \sum_{k=1}^{n} a_{ik} \times b_{kj} \]

Where:

\( c_{ij} \) — Element in the resulting matrix at position (i,j)
\( a_{ik} \) — Element from matrix A at position (i,k)
\( b_{kj} \) — Element from matrix B at position (k,j)
\( n \) — Number of columns in matrix A (must equal number of rows in matrix B)

Explanation: Each element in 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 various fields including computer graphics, physics simulations, economics, engineering, and machine learning algorithms.

4. Using the Calculator

Tips: Enter all elements of the 2x3 matrix A and 3x2 matrix B. The calculator will compute the resulting 2x2 matrix C. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: Why must the inner dimensions match for matrix multiplication?
A: The number of columns in the first matrix must equal the number of rows in the second matrix because each element in the result is computed as the sum of products of corresponding elements.

Q2: Is matrix multiplication commutative?
A: No, matrix multiplication is not commutative. In general, A×B ≠ B×A, and sometimes one product may be defined while the other is not.

Q3: What is the resulting matrix size?
A: When multiplying an m×n matrix by an n×p matrix, the result is an m×p matrix. Here, 2×3 × 3×2 gives a 2×2 matrix.

Q4: Can I multiply matrices of other sizes?
A: This calculator is specifically designed for 2×3 and 3×2 matrices. For other sizes, you would need a different calculator.

Q5: What are some real-world applications of 2x3 and 3x2 matrix multiplication?
A: These matrix sizes are commonly used in transformations between 2D and 3D coordinate systems, simple linear transformations, and various engineering applications.