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LIN-23

Cramer x

2×2 Cramer x.

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GoverningCramer x

Governing equation

x=det(Ax)/detAx=\\det(A_x)/\\det A

e∈[−20,20]e \in \left[-20, 20\right]

where

a
a (—)
b
b (—)
c
c (—)
d
d (—)
e
e (—)
f
f (—)
x
Cramer x (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

LIN-23 · curve
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①

Governing law — red

x=det(Ax)/detAx=\\det(A_x)/\\det A

e∈[−20,20]e \in \left[-20, 20\right]

e=−20e = -20x=1.9×1010x = 1.9\times 10^{10}

②

Domain of the variable — blue

emin⁡=−20,emax⁡=20e_{\min} = -20,\quad e_{\max} = 20

e∈[−20,20]e \in \left[-20, 20\right]

e=−20e = -20e=−20e = -20

③

Response curve — green

x=det(Ax)/detAx=\\det(A_x)/\\det A

e∈[−20,20]e \in \left[-20, 20\right]

e=−20e = -20x=1.9×1010x = 1.9\times 10^{10}

④

Rate of change — amber

dxde\frac{\mathrm{d}x}{\mathrm{d}e}

e∈[−20,20]e \in \left[-20, 20\right]

e=−19.75e = -19.75x′=−1×109x^{\prime} = -1\times 10^{9}

Lecture brief

Historical brief

Determinants, eigenvalues, least squares and Cayley–Hamilton are the nineteenth-century matrix craft behind every coupled system. The sheets invert, project and diagonalise small n. This sheet (LIN-23 — Cramer x) is the form associated with Cramer x. Working symbols: aa, bb, cc, dd, ee, ff →\rightarrow xx. 2×2 Cramer x. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute xx from aa, bb, cc, dd, ee, ff in Linear algebra via x=det(Ax)/detAx=\\det(A_x)/\\det A 2×2 Cramer x. Use it when a real linear algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given a=2.000 —a = 2.000\,\mathrm{—}, b=1.000 —b = 1.000\,\mathrm{—}, c=1.000 —c = 1.000\,\mathrm{—}, d=−1.000 —d = -1.000\,\mathrm{—}, e=5.000 —e = 5.000\,\mathrm{—}, f=1.000 —f = 1.000\,\mathrm{—}, the governing relation x=det(Ax)/detAx=\\det(A_x)/\\det A yields x=−6.000e+9 —x = -6.000e+9\,\mathrm{—}. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Cramer x x-6000000000.000 —
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LIN-23 · curve
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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

2×2 Cramer x. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube