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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.
उदाहरण · rref [[1,2,3],[4,5,6],[7,8,9]]
Rref [[1,2,3],[4,5,6],[7,8,9]]
चरणद्वारा चरण
- \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.
जवाफ प्रकट गर्नुहोस्
अब तपाईँ समस्या रोज्नुहोस्, वा टाइप गर्नुहोस् वा तपाईँको आफ्नै रेखाचित्र गर्नुहोस् । प्रत्येक चरण, एउटा तस्वीर, तपाईँले सोध्नु अघि उत्तर लुकेको छ ।
एक निःशुल्क खाता प्रत्येक पाठ मा नोट थप्छ, तपाईं के समाप्त गरेको छ को एक रेकर्ड, एक ठाउँमा आफ्नो समाधान समस्या, र एक शिक्षक तपाईं यो पृष्ठ बारेमा सोध्न सक्नुहुन्छ. गणित आफैलाई सबैलाई खुला छ, मा साइन इन वा छैन.
साइन अप गर्नुहोस् लगइनयहाँ प्रयोग गरिएको प्रतीक
पूर्ण परिभाषा, एउटा तस्वीर, र यो प्रत्येक अक्षर अर्थ के लागि कुनै पनि प्रतीक ट्याप गर्नुहोस्।
कसरी: 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.
प्रश्नहरू मानिसहरूले सोध्छन्
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.
यसमा थप Linear Algebra
DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis