LIN-01

2×2 determinant

Area of parallelogram.

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Lineardet 2

Governing equation

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

where

a: a (—)
b: b (—)
c: c (—)
d: d (—)
\det: det (—)

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

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

$a = -10$ $\det = -42$

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

ad-bc

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

$a = -10$ $\det = -42$

Along the curve

④

Rate of change — amber

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

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

$a = -9.874$ $\det^{\prime} = 4$

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-01 — 2×2 determinant) is the form associated with det 2. Working symbols: $a$ , $b$ , $c$ , $d$ $\rightarrow$ $\det$ . Area of parallelogram.

Purpose

Purpose: compute

\det

from

a

b

c

d

in Linear algebra via

ad-bc

Area of parallelogram. 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 = 2.000\,\mathrm{—}

d = 4.000\,\mathrm{—}

, the governing relation

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yields

\det = 10.0000\,\mathrm{—}

. Area of parallelogram. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a3 —b b1 —c c2 —d d4 —

Outputs

det \det10.0000 —

Reading speed

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Narration of this film

Area of parallelogram.

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