maths.free › Linear Algebra › Determinants

Determinants

2×2 by ad − bc, larger by cofactor expansion, and what the number means.

The determinant of a matrix is the factor by which it scales area (2×2) or volume (3×3), with a sign for orientation. A zero determinant means the matrix squashes space flat and cannot be inverted. The 3D view draws the column vectors and the box they span.

Eksempel på arbeid · det [[1,2],[3,4]]

Linear-Algebra · Determinants

Det [[1,2],[3,4]]

\left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]

Steg for trinn

  1. \det\left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]

    Determinant of a 2×2 matrix.

  2. = (1)(4) - (2)(3)

    For 2×2: ad − bc.

  3. \det A = -2

    Result.

Vis svaret
\det A = -2

Nå har du Velg et problem, eller skriv inn, eller tegne din egen. Hvert steg, et bilde, svaret skjult til du spør.

Fortsett ditt eget arbeid

En gratis konto legger til notater om hver leksjon, en oversikt over hva du har fullført, problemene du har løst på ett sted og en lærer du kan spørre om. Matematikkene er åpne for alle, signerte på eller ikke.

Registrer Logg inn

Symboler som brukes her

Trykk på et symbol for den fulle definisjonen, et bilde og hva hver bokstav i det betyr.

Hvordan: Determinants

  1. Determinant of a 2×2 matrix.
  2. For 2×2: ad − bc.
  3. Result.

Spørsmål folk stiller

What does a determinant mean geometrically?

It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.

What is an eigenvector?

A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.

Why is matrix multiplication not commutative?

Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.

Mer i Linear Algebra

Matrix inverseEigenvalues and eigenvectorsMatrix multiplicationRow reductionVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis