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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 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]

Àwọn ìṣàmúlò-ètò

\left[\begin{matrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{matrix}\right]
Gauss-Jordan elimination.
\left[\begin{matrix}1 & 2 & 3\\0 & -3 & -6\\7 & 8 & 9\end{matrix}\right]
R2 ← R2 − (4)·R1 to clear column 1.
\left[\begin{matrix}1 & 2 & 3\\0 & -3 & -6\\0 & -6 & -12\end{matrix}\right]
R3 ← R3 − (7)·R1 to clear column 1.
\left[\begin{matrix}1 & 2 & 3\\0 & 1 & 2\\0 & -6 & -12\end{matrix}\right]
R2 ← R2 / -3 to make the pivot 1.
\left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & -6 & -12\end{matrix}\right]
R1 ← R1 − (2)·R2 to clear column 2.
\left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & 0 & 0\end{matrix}\right]
R3 ← R3 − (-6)·R2 to clear column 2.
\left[\begin{matrix}1 & 0 & -1\\0 & 1 & 2\\0 & 0 & 0\end{matrix}\right]
Reduced row-echelon form.

Fi àwọn àgbèwọlé hàn

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

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Bii o ṣe le: Row reduction

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

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

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