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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.

Pracovný príklad: det [[1,2],[3,4]]

Linear-Algebra · Determinants

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

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

Krok za krokom

  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.

Odhaliť odpoveď
\det A = -2

Symbols used here

\det A,\ |A|
determinant
Scaling factor of area/volume under A; zero means singular.
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
matrix
A rectangular array of numbers; a linear map.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.
A^{-1},\ A^{T}
inverse, transpose
The matrix that undoes A; A with rows and columns swapped.
\lambda
lambda (eigenvalue)
The factor by which an eigenvector is stretched: Av = λv.
\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|
dot product, norm
Σ u_i v_i; the length of v, √(v·v).

How to: Determinants

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

Questions people ask

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.

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Viac v kategórii 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