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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.

Mfano wenye matokeo · det [[1,2],[3,4]]

Linear-Algebra · Determinants

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

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

Hatua kwa hatua

  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.

Lafunua jibu
\det A = -2

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Jinsi ya Kufaulu: Determinants

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

Maswali ambayo watu huuliza

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.

Mengi zaidi katika 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