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Linear System Solver (2×2)

Solve coupled linear equations Ax = b from circuit and mechanics models

Parameters

ⓘ
ⓘ
ⓘ
ⓘ
ⓘ
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Controls

xⓘ

Calculated Values

x₁:
1.40;1.40;
x₂:
1.20;1.20;
det(A):
5.00;5.00;
Residual |Ax−b|:
0.00;0.00;

Examples

Sample system

[[2,1],[1,3]] x = [4,5].

  • x1: 1.401.40
  • x2: 1.201.20

Visualization

Linear Systems in Physics

Many steady-state physics problems reduce to Ax = b: two-loop Kirchhoff laws, coupled spring equilibria, mixing constraints, and network flows. For 2×2 systems, Cramer’s rule and Gaussian elimination are fast and instructive.

The determinant det(A) = a₁₁a₂₂ − a₁₂a₂₁ measures whether the system has a unique solution. det(A) = 0 means the equations are parallel or redundant — physically, the model is under- or over-constrained.

Gaussian elimination subtracts multiples of one equation from another to reach upper-triangular form, then back-substitutes. This scales to large sparse systems in simulation codes.

Always verify by substituting x back into Ax = b. Residuals |Ax − b| reveal round-off or modeling errors — propagate uncertainties with Error Propagation on measured coefficients.

Eigenvalue problems (Power Method) extend linear algebra to normal modes and stability analysis.

Key Concepts

  • det(A) ≠ 0 → unique solution
  • Cramer: xᵢ = det(Aᵢ)/det(A)
  • Elimination: pivot, eliminate, back-substitute
  • Residual r = b − Ax checks accuracy
  • Ill-conditioned matrices amplify errors

Real-World Applications

  • Two-loop DC circuit currents
  • Static equilibrium of linked bodies
  • Least-squares normal equations (2-parameter fit)
  • Small vibration modes (with eigen tools)

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Physics Equations

System:
(a11a12a21a22)(x1x2)=(b1b2)\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}\begin{pmatrix}x_1\\x_2\end{pmatrix}=\begin{pmatrix}b_1\\b_2\end{pmatrix}
det(A):
det⁡(A)=a11a22−a12a21\det(A)=a_{11}a_{22}-a_{12}a_{21}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Write the 2×2 system

Coupled linear equations from circuit loops, force balance, or network models.

Equation:

(2113)(x1x2)=(45)\begin{pmatrix}2 & 1\\1 & 3\end{pmatrix}\begin{pmatrix}x_1\\x_2\end{pmatrix}=\begin{pmatrix}4\\5\end{pmatrix}
2

Step 2: Determinant

Equation:

det⁡(A)=a11a22−a12a21\det(A) = a_{11}a_{22} - a_{12}a_{21}

Calculation:

det⁡(A)=2×3−1×1=5.000000\det(A) = 2\times3 - 1\times1 = 5.000000

Result:

Nonsingular system
3

Step 3: Cramer / elimination solution

Equation:

x1=b1a22−b2a12det⁡(A),x2=a11b2−a21b1det⁡(A)x_1 = \frac{b_1 a_{22} - b_2 a_{12}}{\det(A)},\quad x_2 = \frac{a_{11} b_2 - a_{21} b_1}{\det(A)}

Calculation:

x1=1.400000,x2=1.200000x_1 = 1.400000,\quad x_2 = 1.200000

Result:

Solution: (1.4000, 1.2000)
4

Step 4: Verify A x = b

Calculation:

a11x1+a12x2=4.0000,a21x1+a22x2=5.0000a_{11}x_1+a_{12}x_2 = 4.0000,\quad a_{21}x_1+a_{22}x_2 = 5.0000

Result:

Shouldmatchb1=4,b2=5Should match b₁ = 4, b₂ = 5

Frequently Asked Questions (FAQ)

What if det(A) = 0?

No unique solution — revise the physical model or remove redundancy.

Does this scale to 3×3?

Same elimination idea; use matrix tools for larger n.

Practice MCQs

  1. det(A) = 0 implies:
  2. Gaussian elimination goal:
  3. Residual b − Ax should be:
  4. 2×2 Cramer uses:
  5. Coupled springs at equilibrium give:

Related Calculators

These tools connect to the same physics concepts used in this calculator.

Power MethodDominant eigenvalue by iteration.Ohm's LawCombine measured V, I, R with propagated uncertainties.Multiple loop circuits lead to coupled linear equations for currents.Error PropagationCombine uncertainties when multiplying measured quantities.Numerical IntegrationTrapezoidal and Simpson rules for areas under physics curves.Gradient DescentIteratively minimize f(x) by following the negative gradient with animated path visualization.1D Heat EquationFinite-difference FTCS solution to the diffusion equation with animated temperature profiles.1D Wave EquationLeapfrog finite-difference solution to the wave equation with animated wave propagation.Maxwell-Boltzmann DistributionCalculate velocity distributions and characteristic velocities for ideal gases using Maxwell-Boltzmann statistics.Related Statistical Physics topic.