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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.
ตัวอย่างที่ทำงานแล้ว · rref [[1,2,3],[4,5,6],[7,8,9]]
Rref [[1,2,3],[4,5,6],[7,8,9]]
ก้าวต่อก้าว
- \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.
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วิธีการ: 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.
คำถามที่คนถาม
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.
เพิ่มเติมใน Linear Algebra
DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis