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Row reduction

Gaussian elimination, rank and solving Ax = b.

Three row operations — swap, scale, add a multiple of one row to another — bring any matrix to reduced row-echelon form. The number of pivots is the rank; the reduced form of an augmented matrix reads out the solution of a linear system.

Umzekelo osebenzelayo: rref [[1,2,3],[4,5,6],[7,8,9]]

Linear-Algebra · Row-Reduction

Rref [[1,2,3],[4,5,6],[7,8,9]]

\left[\begin{matrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{matrix}\right]

Inyathelo ngenyathelo

  1. \left[\begin{matrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{matrix}\right]

    Gauss–Jordan elimination.

  2. \left[\begin{matrix}1 & 2 & 3\\0 & -3 & -6\\7 & 8 & 9\end{matrix}\right]

    R2 ← R2 − (4)·R1 to clear column 1.

  3. \left[\begin{matrix}1 & 2 & 3\\0 & -3 & -6\\0 & -6 & -12\end{matrix}\right]

    R3 ← R3 − (7)·R1 to clear column 1.

  4. \left[\begin{matrix}1 & 2 & 3\\0 & 1 & 2\\0 & -6 & -12\end{matrix}\right]

    R2 ← R2 / -3 to make the pivot 1.

  5. \left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & -6 & -12\end{matrix}\right]

    R1 ← R1 − (2)·R2 to clear column 2.

  6. \left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & 0 & 0\end{matrix}\right]

    R3 ← R3 − (-6)·R2 to clear column 2.

  7. \left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & 0 & 0\end{matrix}\right]

    Reduced row-echelon form.

Bonisa impendulo
\left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & 0 & 0\end{matrix}\right]

Symbols used here

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.
\det A,\ |A|
determinant
Scaling factor of area/volume under A; zero means singular.
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: Row reduction

  1. Gauss–Jordan elimination.
  2. R2 ← R2 − (4)·R1 to clear column 1.
  3. R3 ← R3 − (7)·R1 to clear column 1.
  4. R2 ← R2 / -3 to make the pivot 1.
  5. R1 ← R1 − (2)·R2 to clear column 2.
  6. R3 ← R3 − (-6)·R2 to clear column 2.
  7. Reduced row-echelon form.

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.

Zama ngokwakho

IiNkqubo Linear Algebra

DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis