matrix multiplication calculator

3×3 Matrix Multiplication Calculator

Enter the values for Matrix A and Matrix B below to calculate their product (Matrix C).

Please ensure all matrix cells contain valid numbers.

Matrix A

×

Matrix B

=

Result (Matrix C)

Understanding Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra used heavily in physics, computer graphics, engineering, and data science. Unlike standard element-wise multiplication, multiplying matrices involves a specific "row-by-column" procedure.

The Rules of Matrix Multiplication

To multiply two matrices, Matrix A and Matrix B, they must be compatible. The number of **columns in Matrix A** must equal the number of **rows in Matrix B**.

If Matrix A has dimensions m × n and Matrix B has dimensions n × p, the resulting Matrix C will have dimensions m × p. This calculator specifically handles 3x3 matrices multiplied by 3x3 matrices, resulting in another 3x3 matrix.

The Calculation Process (Dot Product)

Each element in the resulting matrix is calculated by taking the "dot product" of a row from the first matrix and a column from the second matrix.

To find the value located at Row 1, Column 1 of the result (C_1,1), you multiply the elements of Row 1 of Matrix A by the corresponding elements of Column 1 of Matrix B and sum them up:

C_1,1 = (A_1,1 × B_1,1) + (A_1,2 × B_2,1) + (A_1,3 × B_3,1)

This process is repeated for every cell in the resulting matrix. To find C_2,3, you would calculate the dot product of Row 2 of Matrix A and Column 3 of Matrix B.

Example Calculation

Let's use the default values provided in the calculator above:

Row 1 of A: [1, 2, 3]
Column 1 of B: [9, 6, 3]

To find the top-left result (C_1,1):
(1 × 9) + (2 × 6) + (3 × 3)
= 9 + 12 + 9
= 30

You can verify this result using the calculator provided above.