High School Matrix Multiplication Calculator
Multiply two matrices ($A \times B$) to find the product matrix. Visualize the row-by-column dot product process instantly.
Dot Product Visualizer
Blue rows from A multiply with Red columns from B to create Green cells in C.
Understanding Matrix Multiplication
In advanced algebra and computer graphics, matrices are the engines of transformation. However, multiplying them isn’t as simple as multiplying regular numbers. You calculate the product using a “Row by Column” method, essentially taking the dot product of rows from the first matrix and columns from the second. This Matrix Multiplication Calculator () handles the heavy lifting, allowing you to focus on the logic rather than the arithmetic.
The Dot Product Rule
To find the entry in the $i$-th row and $j$-th column of the result matrix, you multiply the entire $i$-th row of Matrix A by the entire $j$-th column of Matrix B and sum the results.
Specifically, this means the first number in the row hits the first number in the column, the second hits the second, and so on. Then, you add all those products together to get a single number for that spot in the result.
Why Dimensions Matter
Furthermore, you cannot multiply just any two matrices. The number of columns in the first matrix ($n$) must exactly match the number of rows in the second matrix ($n$). Consequently, the resulting matrix will have dimensions $m \times p$ (Rows of A by Cols of B). Using a Matrix Multiplication Calculator () helps you visualize these dimensional constraints before you start calculating.
Frequently Asked Questions
Generally, No. In standard numbers, $2 \times 3$ is the same as $3 \times 2$. In contrast, with matrices, $A \times B$ is rarely the same as $B \times A$. In fact, if the dimensions don’t align, $B \times A$ might not even be possible!
Manual multiplication involves many small calculations (a 3×3 multiplication requires 27 multiplications and 18 additions). Therefore, this tool is invaluable for checking your work and avoiding simple arithmetic errors during homework.
The Identity Matrix ($I$) acts like the number “1”. Any matrix $A$ multiplied by $I$ results in $A$ unchanged. It is a square matrix with 1s on the diagonal and 0s everywhere else.
We hope this Matrix Multiplication Calculator () helps you ace your linear algebra exams!