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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · det [[1,2],[3,4]]

Linear-Algebra · Determinants

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

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

Àwọn ìṣàmúlò-ètò

\det\left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]
Determinant of a 2×2 matrix.
= (1)(4) - (2)(3)
For 2×2: ad − bc.
\det A = -2
Result.

Fi àwọn àgbèwọlé hàn

\det A = -2

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Bii o ṣe le: Determinants

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

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Linear Algebra

Matrix inverse Eigenvalues and eigenvectors Matrix multiplication Row reduction Vector spaces, span and linear independence Orthogonality, projections and least squares Diagonalisation and matrix powers Linear transformations and change of basis