Everyday Tools

Matrix-Vector Multiplication Calculator

Multiply a small matrix by a compatible vector in practical 2x2 and 3x3 modes.

Updated April 17, 2026
Free online tool
Planning and research use

Why this page exists

Linear-algebra work is easier to check when a matrix-vector product is calculated directly instead of being rebuilt row by row by hand every time. This calculator helps users multiply a small matrix by a compatible vector in 2x2 and 3x3 modes and clearly shows the original matrix, the original vector, and the resulting vector.

Interactive tool

Run the estimate

Enter your numbers and read the result first, then use the sections below to understand what affects the outcome.

Calculator

Matrix-vector multiplication calculator

Multiply a small matrix by a compatible vector in student-friendly 2x2 and 3x3 modes.

Result

<5, 8, 17>

Calculated the matrix-vector product by multiplying each matrix row by the compatible input vector.

Resulting vector<5, 8, 17>

Original matrix used[[1, 2, 0], [-1, 3, 4], [2, 1, 5]]

Original vector used<3, 1, 2>

Mode used3x3 matrix × 3D vector

3x3 matrix multiplication with <3, 1, 2> gives <5, 8, 17> in this matrix-vector estimate.
Each result entry comes from one matrix row multiplied component by component with the compatible input vector, then summed.
Use the result with matrix and vector tools if you want to compare the transformation with broader linear-algebra work.

This is standard matrix-vector multiplication for the supported small modes only. The matrix and vector sizes must stay compatible with the mode selected.

Planning note

Last updated April 17, 2026. Use this tool to compare scenarios and plan ahead, then confirm important details with the lender, employer, insurer, contractor, or other qualified provider involved in the final decision.

How it works

What the calculator is doing

Choose the supported matrix-vector mode you want to use and enter the matrix entries and vector components.

The calculator multiplies each matrix row by the compatible vector and sums the row products to build the resulting vector.

It shows the product vector together with the original matrix, vector, and mode used.

Understanding your result

This is standard matrix-vector multiplication for the supported small modes only. The matrix and vector must stay dimensionally compatible with the selected mode.

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Examples

Ways people use this tool

Example scenarios help turn a quick estimate into a more useful comparison or planning step.

Check a matrix transformation homework step

A direct result can help confirm whether each row-product sum was handled correctly.

Compare 2x2 and 3x3 transformation outputs

Switching modes can show how a different matrix size changes the resulting vector format and outcome.

Use it with matrix and vector tools

Matrix-vector multiplication becomes more useful when reviewed beside other matrix and vector operations in the same workflow.

When to use it

Good times to run this calculator

Use this when you want a quick check on a small matrix-vector product in a homework, physics, or linear-algebra problem.

It is especially useful when you want to verify row-by-row multiplication without rebuilding every sum manually.

Assumptions and limitations

The calculator supports only the practical 2x2-by-2D and 3x3-by-3D modes shown on the page.

It does not handle symbolic entries, larger matrices, or incompatible dimensions outside the supported modes.

Common mistakes

Avoid the usual input mistakes

Mixing the order of vector components can change every row sum even when the matrix entries themselves are correct.

Confusing matrix-vector multiplication with full matrix-matrix multiplication can lead to the wrong expected output shape.

Practical tips

Use the matrix label and vector label shown in the result to double-check that you entered values in the intended order before trusting the product.

Pair the product with vector-angle or magnitude tools if you want to interpret how the transformation changes direction or size.

Worked example

Walk through a realistic scenario

A worked example shows how the estimate behaves when the inputs resemble a real planning decision.

Multiply a 3x3 matrix by a 3D vector

A student wants to apply a 3x3 matrix to a 3D vector and confirm the resulting transformed vector without redoing every row sum by hand.

1. Choose the 3x3 mode and enter the matrix entries and vector components.

2. Multiply each matrix row by the vector component by component.

3. Add each row’s products to read the resulting vector.

Takeaway: The result gives a clean transformation check and makes the compatible dimensions easy to confirm.

FAQ

Common questions

How is matrix-vector multiplication calculated here?

The calculator multiplies each matrix row by the compatible vector component by component and then sums those products to produce each entry in the resulting vector.

Why do the dimensions have to match the selected mode?

Because each row needs the same number of components as the vector, so a 2x2 matrix must pair with a 2D vector and a 3x3 matrix must pair with a 3D vector.

Why is this different from matrix-matrix multiplication?

Matrix-vector multiplication uses one vector as the input on the right side, so the output is a vector rather than another matrix.

Related tools

Keep comparing

Matrix multiplication, addition, subtraction, and vector-magnitude tools help show how the matrix-vector product fits inside broader linear-algebra work.

Vector-angle and dot-product tools add context when you want to interpret how the transformed vector compares directionally with other vectors.

Everyday ToolsUpdated April 16, 2026