Find The Product Of Two Matrices Calculator

Easily calculate the resultant matrix from the multiplication of two matrices with our online product of two matrices calculator.

Matrix Multiplication Calculator

Matrix A:

Matrix B:

Result Matrix C:

Result Matrix C (A x B):

Formula: C[i][j] = Σ (A[i][k] * B[k][j])

What is the Product of Two Matrices?

The product of two matrices is a fundamental operation in linear algebra. When you multiply two matrices, say matrix A and matrix B, the result is a new matrix, say C. However, matrix multiplication is only defined if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B). If matrix A is of size m x n (m rows, n columns) and matrix B is of size n x p (n rows, p columns), their product C will be a matrix of size m x p.

Each element C_ij (element in the i-th row and j-th column of C) is calculated by taking the dot product of the i-th row of A and the j-th column of B. This means you multiply corresponding elements from the row and column and sum the results. Our product of two matrices calculator automates this process.

This operation is widely used in various fields like computer graphics (for transformations like scaling, rotation, and translation), physics (in quantum mechanics and optics), engineering, data analysis, and machine learning. Anyone working with linear transformations or systems of linear equations will likely use a product of two matrices calculator or perform this operation.

A common misconception is that matrix multiplication is commutative (i.e., A x B = B x A). This is generally NOT true. The order of multiplication matters significantly.

Product of Two Matrices Formula and Mathematical Explanation

If A is an m x n matrix and B is an n x p matrix, their product C = A x B is an m x p matrix where each element C_ij is given by the formula:

C_ij = ∑_k=1ⁿ A_ik * B_kj

This means: C_ij = A_i1B_1j + A_i2B_2j + … + A_inB_nj

For example, to find the element in the first row and first column of the product matrix C (C₁₁), you take the elements of the first row of A and the elements of the first column of B, multiply them pair-wise, and sum the results.

Our product of two matrices calculator implements this formula precisely.

Variable	Meaning	Context
Aik	Element in the i-th row and k-th column of matrix A	Matrix A elements
Bkj	Element in the k-th row and j-th column of matrix B	Matrix B elements
Cij	Element in the i-th row and j-th column of the product matrix C	Result matrix elements
m	Number of rows in matrix A	Dimension
n	Number of columns in A / rows in B	Dimension
p	Number of columns in matrix B	Dimension

Practical Examples (Real-World Use Cases)

Using a product of two matrices calculator is common in many fields.

Example 1: Simple 2×2 Matrix Multiplication

Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]

Matrix A is 2×2, Matrix B is 2×2. The product C will be 2×2.

C₁₁ = (1 * 5) + (2 * 7) = 5 + 14 = 19

C₁₂ = (1 * 6) + (2 * 8) = 6 + 16 = 22

C₂₁ = (3 * 5) + (4 * 7) = 15 + 28 = 43

C₂₂ = (3 * 6) + (4 * 8) = 18 + 32 = 50

So, C = [[19, 22], [43, 50]]. Our product of two matrices calculator would give this result.

Example 2: Different Dimensions

Let A = [[1, 0, 2], [-1, 3, 1]] (2×3 matrix) and B = [[3, 1], [2, 1], [1, 0]] (3×2 matrix)

The product C will be a 2×2 matrix.

C₁₁ = (1*3) + (0*2) + (2*1) = 3 + 0 + 2 = 5

C₁₂ = (1*1) + (0*1) + (2*0) = 1 + 0 + 0 = 1

C₂₁ = (-1*3) + (3*2) + (1*1) = -3 + 6 + 1 = 4

C₂₂ = (-1*1) + (3*1) + (1*0) = -1 + 3 + 0 = 2

So, C = [[5, 1], [4, 2]]. You can verify this with the product of two matrices calculator.

How to Use This Product of Two Matrices Calculator

1. Enter Dimensions: Specify the number of rows for Matrix A, the number of columns for Matrix A (which is also the number of rows for Matrix B), and the number of columns for Matrix B using the input fields. The maximum allowed is 4 for each dimension.
2. Generate Inputs: Click “Set Dimensions & Generate Inputs”. The calculator will create input fields for the elements of Matrix A and Matrix B based on the dimensions you provided.
3. Enter Matrix Elements: Fill in the numerical values for each element of Matrix A and Matrix B in the generated input grids.
4. Calculate: Click the “Calculate Product” button.
5. View Results: The product of two matrices calculator will display the resulting Matrix C below the buttons. It will also show the formula and, if applicable, intermediate calculation details for C[0][0] and a chart.
6. Reset: Click “Reset” to clear all inputs and start over with default 2×2 matrices.
7. Copy Results: Click “Copy Results” to copy the input matrices and the result matrix to your clipboard.

The chart visualizes the terms (A[0][k] * B[k][0]) that are summed to get the element C[0][0] of the result matrix, helping to understand the calculation of the first element.

Key Factors That Affect Product of Two Matrices Results

• Matrix Dimensions: The number of rows and columns (m, n, p) directly determines the size of the resulting matrix and whether multiplication is even possible (n must be the same).
• Element Values: The individual numerical values within each matrix are the core components of the calculation. Small changes in these values can lead to significant changes in the product matrix.
• Order of Multiplication: A x B is generally not equal to B x A. The order in which you multiply the matrices is crucial.
• Zero Elements: The presence of zeros can simplify calculations but also affect the magnitude of the results.
• Negative Values: Negative numbers in the matrices will influence the signs and values of the elements in the product matrix.
• Computational Precision: For very large numbers or many decimal places, the precision of the calculation can matter, though our product of two matrices calculator uses standard JavaScript precision.

Frequently Asked Questions (FAQ)

Q1: When is matrix multiplication possible?

A1: Matrix multiplication of A x B is only possible if the number of columns in matrix A is equal to the number of rows in matrix B.

Q2: Is matrix multiplication commutative?

A2: No, in general, A x B ≠ B x A. The order matters.

Q3: What is the size of the resulting matrix?

A3: If A is m x n and B is n x p, the product C = A x B will be an m x p matrix.

Q4: Can I multiply a matrix by a scalar using this calculator?

A4: This product of two matrices calculator is designed for multiplying two matrices. Scalar multiplication involves multiplying every element of a matrix by a single number, which is a different operation.

Q5: What if one of my matrices is just a row or column vector?

A5: A row vector is a 1 x n matrix, and a column vector is an m x 1 matrix. You can use the calculator by setting the appropriate dimensions (e.g., 1 row for a row vector).

Q6: How are the elements of the product matrix calculated?

A6: Each element C_ij is the dot product of the i-th row of A and the j-th column of B.

Q7: What is the identity matrix in matrix multiplication?

A7: The identity matrix (I) is a square matrix with 1s on the main diagonal and 0s elsewhere. It acts like the number 1 in multiplication: A x I = A and I x A = A (if dimensions match).

Q8: What happens if I enter non-numeric values?

A8: The calculator expects numeric values. Non-numeric input will likely result in NaN (Not a Number) or errors in the calculation. The calculator attempts to parse numbers and will show 0 or NaN if input is invalid.