LIN-02

2×2 inverse det

Guard det=0.

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Linearinverse 2

Governing equation

A^{-1}=(1/\\det)\\mathrm{adj}

$a \in \left[-10, 10\right]$

where

a: a (—)
b: b (—)
c: c (—)
d: d (—)
\det: det (—)
A^{-1}_{11}: a' (—)

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.

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①

Governing law — red

A^{-1}=(1/\\det)\\mathrm{adj}

$a \in \left[-10, 10\right]$

$a = -10$ $\det = -20$

Along the curve

②

Domain of the variable — blue

a_{\min} = -10,\quad a_{\max} = 10

$a \in \left[-10, 10\right]$

$a = -10$ $a = -10$

Along the curve

③

Response curve — green

A^{-1}=(1/\\det)\\mathrm{adj}

$a \in \left[-10, 10\right]$

$a = -10$ $\det = -20$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\det}{\mathrm{d}a}

$a \in \left[-10, 10\right]$

$a = -9.874$ $\det^{\prime} = 2$

Along the curve

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-02 — 2×2 inverse det) is the form associated with inverse 2. Working symbols: $a$ , $b$ , $c$ , $d$ $\rightarrow$ $\det$ , $A^{-1}_{11}$ . Guard det=0.

Purpose

Purpose: compute

\det

A^{-1}_{11}

from

a

b

c

d

in Linear algebra via

A^{-1}=(1/\\det)\\mathrm{adj}

Guard det=0. 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 = 3.000\,\mathrm{—}

b = 1.000\,\mathrm{—}

c = 0.000\,\mathrm{—}

d = 2.000\,\mathrm{—}

, the governing relation

A^{-1}=(1/\\det)\\mathrm{adj}

yields

\det = 6.0000\,\mathrm{—}

A^{-1}_{11} = 0.3333\,\mathrm{—}

. Guard det=0. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a3 —b b1 —c c0 —d d2 —

Outputs

det \det6.0000 —
a' A^{-1}_{11}0.3333 —

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Narration of this film

Guard det=0.

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