Find The Product Of Matrices Calculator

Enter the dimensions and elements of two matrices to calculate their product. The number of columns in Matrix A must equal the number of rows in Matrix B.

Matrix A Elements:

Matrix B Elements:

What is the Product of Matrices?

The Product of Matrices, also known as matrix multiplication, is a binary operation that produces a single matrix from two matrices. For the multiplication to be defined, the number of columns in the first matrix (Matrix A) must be equal to the number of rows in the second matrix (Matrix B). If Matrix A is an m × n matrix and Matrix B is an n × p matrix, their product AB will be an m × p matrix.

This operation is fundamental in linear algebra and has wide applications in various fields like physics, engineering, computer graphics, economics, and data science. Unlike scalar multiplication (multiplying a matrix by a number), the Product of Matrices is generally not commutative, meaning AB is not always equal to BA.

Who should use it? Anyone working with systems of linear equations, transformations of geometric objects, data analysis, or simulations involving vector spaces will frequently encounter the need to calculate the Product of Matrices.

Common misconceptions include thinking matrix multiplication is element-wise (like addition) or that it's always commutative.

Product of Matrices Formula and Mathematical Explanation

If A is an m × n matrix and B is an n × p matrix, their product C = AB is an m × p matrix, where each element C_ij (in the i-th row and j-th column of C) is calculated as:

C_ij = A_i1B_1j + A_i2B_2j + … + A_inB_nj = Σ_k=1ⁿ A_ikB_kj

This means each element of the resulting matrix is the dot product of a row from the first matrix and a column from the second matrix.

Variables Table

Variables involved in calculating the Product of Matrices.

Variable	Meaning	Unit	Typical Range
Aik	Element in the i-th row and k-th column of Matrix A	Dimensionless (or unit of underlying data)	Real or complex numbers
Bkj	Element in the k-th row and j-th column of Matrix B	Dimensionless (or unit of underlying data)	Real or complex numbers
Cij	Element in the i-th row and j-th column of the Product Matrix C	Dimensionless (or unit of underlying data)	Real or complex numbers
m	Number of rows in Matrix A (and C)	Integer	1, 2, 3, …
n	Number of columns in Matrix A / rows in Matrix B	Integer	1, 2, 3, …
p	Number of columns in Matrix B (and C)	Integer	1, 2, 3, …

Practical Examples (Real-World Use Cases)

Example 1: Transforming Coordinates in Computer Graphics

Suppose we have a point (x, y) represented as a 1×2 matrix [x y] and we want to rotate it by an angle θ counterclockwise. The rotation matrix is a 2×2 matrix:

R = [[cos(θ), sin(θ)], [-sin(θ), cos(θ)]]

Let's say the point is (2, 3) and the angle is 90 degrees (cos(90)=0, sin(90)=1). So, A = [2 3] (1×2) and R = [[0, 1], [-1, 0]] (2×2). The product is not [2 3] * R, but rather R * [2; 3] (a 2×1 column vector). Or, if we use row vectors, we multiply by the transpose of R. Let's use column vector for the point [2; 3] (2×1) and R (2×2) as defined.

A (point) = [[2], [3]] (2×1), R = [[0, 1], [-1, 0]] (2×2). Product R * A:

R * A = [[0*2 + 1*3], [-1*2 + 0*3]] = [[3], [-2]]

The new point is (3, -2). Here, the number of columns of R (2) matches the number of rows of A (2). The result is 2×1.

Example 2: Systems of Linear Equations

Consider the system:

2x + 3y = 7
1x + 4y = 6

This can be written in matrix form as AX = B, where A = [[2, 3], [1, 4]], X = [[x], [y]], B = [[7], [6]]. If we knew the inverse of A, A^-1, we could find X by X = A^-1B. Calculating A^-1 involves matrix operations, and then finding X is a Product of Matrices (A^-1 and B).

Let's say we have A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. Both are 2×2. The product AB will be 2×2.

AB₁₁ = 1*5 + 2*7 = 5 + 14 = 19
AB₁₂ = 1*6 + 2*8 = 6 + 16 = 22
AB₂₁ = 3*5 + 4*7 = 15 + 28 = 43
AB₂₂ = 3*6 + 4*8 = 18 + 32 = 50

So, AB = [[19, 22], [43, 50]]. Our Product of Matrices calculator can verify this.

How to Use This Product of Matrices Calculator

1. Enter Dimensions: Input the number of rows for Matrix A, the common dimension (columns of A / rows of B), and the number of columns for Matrix B in the respective fields. The calculator supports dimensions up to 5×5.
2. Input Matrix Elements: Based on the dimensions you entered, input fields for the elements of Matrix A and Matrix B will appear. Enter the numeric values for each element.
3. Calculate: Click the "Calculate Product" button.
4. View Results: The calculator will display the resulting product matrix C = AB, along with the input matrices and their dimensions. If the dimensions are incompatible for multiplication, an error message will appear. The Product of Matrices is shown clearly.
5. Reset: Click "Reset" to clear the inputs and results and start with default 2×2 matrices.
6. Copy Results: Click "Copy Results" to copy the main result and intermediate values to your clipboard.

Key Factors That Affect Product of Matrices Results

1. Matrix Dimensions: The number of columns in the first matrix MUST equal the number of rows in the second. If they don't match, the Product of Matrices is undefined.
2. Order of Multiplication: Matrix multiplication is generally NOT commutative (AB ≠ BA). The order matters significantly.
3. Element Values: The specific numerical values within the matrices directly determine the values in the product matrix.
4. Zero Matrices: If one of the matrices is a zero matrix (all elements are zero), the product will also be a zero matrix (if dimensions allow multiplication).
5. Identity Matrix: Multiplying a matrix by an identity matrix (of compatible size) results in the original matrix (AI = A, IA = A).
6. Singular Matrices: The properties of the matrices (e.g., whether they are singular or invertible) influence the properties of their product, but the calculation method remains the same. The Product of Matrices itself doesn't depend on singularity, but subsequent operations might.

Frequently Asked Questions (FAQ)

What is the condition for matrix multiplication?

The number of columns of the first matrix must be equal to the number of rows of the second matrix.

Is the Product of Matrices commutative?

No, in general, AB is not equal to BA. The order of multiplication matters for the Product of Matrices.

What are the dimensions of the product matrix?

If A is m × n and B is n × p, the product AB is an m × p matrix.

Can I multiply any two matrices?

No, only if their inner dimensions match (columns of first = rows of second).

How is the Product of Matrices different from element-wise multiplication?

Element-wise multiplication (Hadamard product) requires matrices of the same dimensions, and each element of the result is the product of corresponding elements. Matrix multiplication involves row-column dot products.

What happens if I multiply a matrix by an identity matrix?

Multiplying a matrix by a compatible identity matrix results in the original matrix. The Product of Matrices with an identity matrix is straightforward.

What if one matrix has only one row or one column?

These are row or column vectors. Multiplication is still possible if the inner dimensions match, resulting in a matrix, row vector, or column vector.

Where is the Product of Matrices used?

It's used in linear transformations, solving systems of linear equations, computer graphics, quantum mechanics, data analysis, and many other areas of science and engineering. Calculating the Product of Matrices is a core operation.

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