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Matrix Multiplication Calculator

Set matrices' orders and write down their entities to find the product (if possible) up to a 10x10 order through this matrix multiplication calculator.

Matrix A Dimension:

Matrix B Dimension

Matrix A

Matrix B

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Matrix Multiplication Calculator

Use this free matrix multiplication calculator to quickly compute the product of two matrices, whether they contain integers, decimals, or complex numbers. This guide also explains the fundamentals, rules, and properties of matrix multiplication.

What Is a Matrix?

A matrix is a rectangular arrangement of numbers organized in rows and columns, enclosed in brackets [ ]. For example:

$$ \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} $$

Other examples include:

$$ \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix}, \begin{bmatrix} 5 \\ 2 \end{bmatrix}, \begin{bmatrix} 12 \end{bmatrix} $$

General Formula for Matrix Multiplication

Consider two matrices, A and B, where we want to multiply them to get a new matrix C. Each element of the product is calculated as:

$$ A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & \cdots & b_{1p} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{np} \end{bmatrix} $$

$$ C = A \cdot B = \begin{bmatrix} \sum_{k=1}^{n} a_{1k}b_{k1} & \cdots & \sum_{k=1}^{n} a_{1k}b_{kp} \\ \vdots & \ddots & \vdots \\ \sum_{k=1}^{n} a_{mk}b_{k1} & \cdots & \sum_{k=1}^{n} a_{mk}b_{kp} \end{bmatrix} $$

How to Find an Element in the Product Matrix

Select a row from the first matrix and a column from the second matrix.
Multiply corresponding elements of the row and column.
Add all the results together to get the element of the product matrix.

Key Conditions for Matrix Multiplication

The number of columns in the first matrix must match the number of rows in the second matrix.
The resulting matrix will have rows equal to the first matrix and columns equal to the second matrix.
Example: multiplying a matrix of order m × n with another matrix of order n × p produces a matrix of order m × p.

Example:

$$ \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} \cdot \begin{bmatrix} 5 \\ 6 \end{bmatrix} = \begin{bmatrix} 27 \\ 29 \end{bmatrix} $$

Properties of Matrix Multiplication

Commutative: AB ≠ BA
Associative: (AB)C = A(BC)
Distributive: A(B+C) = AB + AC, (A+B)C = AC + BC
Identity: IA = A, AI = A
Zero Matrix: A0 = 0, 0A = 0

Example: Multiplying with the Identity Matrix

Given:

$$ \begin{bmatrix} 7 \\ 3 \end{bmatrix}, \quad I = \begin{bmatrix} 1 & 0 \end{bmatrix} $$

Multiplication:

$$ \begin{bmatrix} 7 \\ 3 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 3 & 0 \end{bmatrix} $$

How the Matrix Multiplication Calculator Works

Specify the number of rows and columns for the first matrix.
Specify the number of rows and columns for the second matrix (rows must match columns of the first matrix).
Click “Set Matrices” to generate input fields.
Enter values for all matrix elements.
Click “Calculate” to get the product matrix along with step-by-step results.

FAQs

How do I multiply 2×2 matrices quickly?

Simply enter the matrices into our free online calculator for instant results.

Can I multiply a 2×3 matrix with a 4×3 matrix?

No. The number of columns in the first matrix must equal the number of rows in the second matrix.

What is the size of the resulting matrix?

The product matrix has rows equal to the first matrix and columns equal to the second matrix.

What is scalar multiplication of a matrix?

Multiply each element of the matrix by a single scalar value.

Other Useful Matrix Tools

Conclusion

Matrix multiplication is a core concept in linear algebra. Using this calculator, you can quickly compute products of any compatible matrices with step-by-step guidance.

References

Wikipedia: Matrix Multiplication – General Properties
Khan Academy: Matrix Transformations
Lumen Learning: Introduction to Matrices

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