Matrix Multiplication: a complete example
A 2×3 matrix times a 3×2 matrix produces a 2×2 matrix. This Matrix Multiplication example can be compared with vector action.
Math calculator
Matrix Multiplication Calculator
Multiply compatible matrices using row-by-column dot products. Formula and matrix product remain visible in one place for independent verification.
Matrix Multiplication inputs
Calculation inputs
Mathematical foundation
A matrix product AB exists when A has as many columns as B has rows; each result entry is a row-column dot product.
Reading matrix product correctly starts with the mathematical structure described here.
Steps without the calculator
Write the output dimensions, pair every A row with every B column, multiply components, and add.
Boundaries and common traps
AB usually differs from BA, and BA may not even be dimensionally defined. If the Matrix Multiplication assumptions do not fit, consider repeated product.
Reversing the displayed steps offers a quick independent check on this matrix multiplication result.
Situations that fit the model
Composition of transformations, linear systems, graphics, networks, and Markov models relies on multiplication.
Reading sensitivity and precision
The input labels—matrix a and matrix b—encode the model used on this page. Write those labels beside source values when transferring a problem from paper or a spreadsheet. That small step catches transposed quantities and mixed units before they become a polished-looking matrix product.
Changing an interior dimension invalidates the product unless the other matrix changes with it. A one-field trial makes this relationship visible without reworking the entire example.
Choose rounding after considering how the result will be used. Comparison may need only a few significant digits, while a later multi-step calculation benefits from carrying more. In either case, retain the page's formula with the matrix product so the underlying definition remains visible.
Matrix-vector multiplication is the one-column special case. The distinction determines whether this page fits the original question.
If the number moves into a spreadsheet, give its cell a matrix product heading and retain the source values matrix a and matrix b nearby. Context matters more than extra displayed digits.
Checking a row-column product
The working inputs for Matrix Multiplication are Matrix A, and Matrix B. Confirm that A columns equal B rows, then recompute one output entry as a row of A dotted with a column of B.
For Matrix Multiplication, record pivot choices and any row swaps when elimination affects the result.
Questions about Matrix Multiplication
Checking What does Matrix Multiplication calculate?
A matrix product AB exists when A has as many columns as B has rows; each result entry is a row-column dot product.
When is Matrix Multiplication useful?
Composition of transformations, linear systems, graphics, networks, and Markov models relies on multiplication.
What can make Matrix Multiplication misleading?
AB usually differs from BA, and BA may not even be dimensionally defined.
Matrix Multiplication independently
Write the output dimensions, pair every A row with every B column, multiply components, and add.