Mathematics · Linear Algebra
2×2 Matrix–Vector Calculator
Apply a 2×2 linear transformation to a two-component vector.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- First component=2×4+0×5=8.
- Second component=0×4+3×5=15.
Understand Matrix–vector product
One idea, three depths
Choose how deeply to explain Matrix–vector product
Matrix–vector product: Apply a 2×2 linear transformation to a two-component vector.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Matrix–vector product to answer this question: apply a 2×2 linear transformation to a two-component vector? Enter A₁₁, A₁₂, A₂₁, and 3 other inputs; the calculator shows Output x. For example: [[2,0],[0,3]] times (4,5) gives (8,15). The answer tells you Output x.
Age 15Explain it to a 15-year-oldConnect it to the formula
Rows of the matrix dot with the input vector to produce transformed components. The rule is y=Ax. Its input values are A₁₁, A₁₂, A₂₁, A₂₂, Vector x, Vector y, and the main result is Output x. For example: [[2,0],[0,3]] times (4,5) gives (8,15).
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated matrix–vector product relation over the valid real-number domain stated below. The implemented relation is y=Ax, evaluated from A₁₁, A₁₂, A₂₁, A₂₂, Vector x, Vector y to produce Output x. Rows of the matrix dot with the input vector to produce transformed components. Vector component order must match the matrix basis.
Inputs and valid domain
- A₁₁ must be a finite real number.
- A₁₂ must be a finite real number.
- A₂₁ must be a finite real number.
- A₂₂ must be a finite real number.
- Vector x must be a finite real number.
- Vector y must be a finite real number.
Important boundary: Vector component order must match the matrix basis.
The formula
y=Ax
How the calculator works through it
It substitutes A₁₁, A₁₂, A₂₁, A₂₂, Vector x, Vector y into the formula and exposes every numerical step above. The main output is Output x, accompanied by Output y, Output magnitude.
Read the result correctly
The Output x is the direct answer to “apply a 2×2 linear transformation to a two-component vector.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
[[2,0],[0,3]] times (4,5) gives (8,15).
Where this model stops being reliable
Vector component order must match the matrix basis.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Matrix–vector product works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Matrix–vector product uses y=Ax. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Vectors and components
Component notation helps you follow how Matrix–vector product combines directional or indexed values.
Review this foundation about 6 min
Optional enrichment
- Matrices and linear transformations
Matrices place Matrix–vector product inside the wider language of linear systems and transformations.
Review this foundation about 7 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read A₁₁, A₁₂, A₂₁, A₂₂, Vector x, Vector y.
- Evaluate the principal relationship: y=Ax.
- Return Output x and check the domain conditions described above.
Python
from math import *
def matrix_vector_2x2(a, b, c, x, x1, y1) -> float:
return ((a * x1) + (b * y1))
assert abs(matrix_vector_2x2(2, 0, 0, 3, 4, 5) - 8) < 1e-6 * max(1.0, abs(8))
C
#include <assert.h>
#include <math.h>
double matrix_vector_2x2(double a, double b, double c, double x, double x1, double y1) {
return ((a * x1) + (b * y1));
}
int main(void) {
const double expected = 8;
const double actual = matrix_vector_2x2(2, 0, 0, 3, 4, 5);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double matrix_vector_2x2(double a, double b, double c, double x, double x1, double y1) {
return ((a * x1) + (b * y1));
}
int main() {
constexpr double expected = 8;
const double actual = matrix_vector_2x2(2, 0, 0, 3, 4, 5);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double matrix_vector_2x2(double a, double b, double c, double x, double x1, double y1)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global matrix_vector_2x2
section .text
matrix_vector_2x2:
push rbp
mov rbp, rsp
sub rsp, 80
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd [rbp-40], xmm4
movsd [rbp-48], xmm5
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-40]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-48]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-64]
addsd xmm0, [rbp-72]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-56]
leave
ret
MATLAB
function result = matrix_vector_2x2(a, b, c, x, x1, y1)
result = ((a * x1) + (b * y1));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, c_, x_, x1_, y1_] := ((a * x1) + (b * y1));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). 2×2 Matrix–Vector Calculator. MW SysArc Tools. https://math.mwsysarc.com/linear-algebra/2x2-matrix-vector
MLA 9
MW SysArc. “2×2 Matrix–Vector Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/linear-algebra/2x2-matrix-vector. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “2×2 Matrix–Vector Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/linear-algebra/2x2-matrix-vector.
Harvard
MW SysArc (2026) ‘2×2 Matrix–Vector Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/linear-algebra/2x2-matrix-vector (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_matrix_vector_2x2_2026,
author = {{MW SysArc}},
title = {2×2 Matrix–Vector Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/linear-algebra/2x2-matrix-vector},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - 2×2 Matrix–Vector Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/linear-algebra/2x2-matrix-vector
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Matrix–vector product do?
Apply a 2×2 linear transformation to a two-component vector.
How does the Matrix–vector product work?
The calculator applies y=Ax. Rows of the matrix dot with the input vector to produce transformed components.
What can I learn from the Matrix–vector product?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .