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https://files.eric.ed.gov/fulltext/EJ1093357.pdf

https://www.scribd.com/doc/238620147/Chio-s-Method

https://www.studocu.com/en-us/document/california-state-university-los-angeles/linear-algebra/chios-method/34530107

Chiò's method

The Chiò method, or more precisely Chiò's pivotal condensation method, is a technique to calculate the determinant of a matrix by reducing its order step-by-step until a 2x2 matrix is obtained. It is not a method for solving a system of equations, but rather a means of evaluating the determinant of a square matrix, which can then be used in other methods like Cramer's rule to solve a system of linear equations. Using integer arithmetic on a computer avoids some sticky issues involving floating point calculations. /p>

Historically, before computers were available, most of examples and exercises in textbooks on Linear Algebra involved integers rather than fractions or irrational numbers or their floating point representations. Obviously, people prefer to keep it that way avoiding heavy manual calculations. Since its appearance in 1853 Chiò's trick has been rediscovered repeatedly, For instance, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author) discovered it in 1866; it is know now as Dodgson condensation or method of contractants.

Felice (Félix) Chiò (1813–1871) was an Italian mathematician. In Italian, his surname is spelled Chiò with a grave accent on the "ò". This is confirmed by Italian academic sources and historical records. However, some references use Chió with an acute accent (é-style). This is likely a typographical or transliteration inconsistency, possibly influenced by Spanish or French conventions.

In 1853 Felice (Félix) Chiò published his short Mémoire sur les fonctions connues sous le noms De Résultantes Ou De Déterminans. The details of Chiò’s life are not well known. In 1846 when he was teaching mathemat- ics at the Military Academy of Turin, he presented a paper to the Academy of Sciences in Paris that was reported in its proceedings, Comptes Rendus (v. XXIII, no. 10, 7 September 1846), and published in 1853 in the form of two memoirs as Recherches sur la série de Lagrange. In 1854 he was appointed to the chair of physics ‘sublime’ at the Military Academy. In the same year he was appointed professor of mathematical physics at the University of Turin.

We demonstrate such approach, which is referred to as Chiò’s method, in some examples. Let us consider a 3 × 3 matrix \[ {\bf A} = \begin{bmatrix} 2& \phantom{-}3& -1 \\ 3& -2& \phantom{-}4 \\ 1& \phantom{-}5& \phantom{-}2 \end{bmatrix} . \] Suppose we want to treat the first entry in the first row as a pivot. In order to eliminate all entries in the first column below the pivot, we multiply by it (in our case, by "2") other rows: \[ {\bf A}_1 = \begin{bmatrix} 2& 3& -1 \\ 6& -4& 8 \\ 2& 10& 4 \end{bmatrix} . \]

A = {{2, 3, -1}, {3, -2, 4}, {1, 5, 2}};
With row reductions R₂ ← R₂ − 3·R₁ and R₃ ← R₃ − R₁, we obtain an equivalent matrix \[ {\bf A}_2 = \begin{bmatrix} 2& 3& -1 \\ 0& -13& 11 \\ 0& 7& 5 \end{bmatrix} . \]
A2 = {A[[1]], 2*A[[2]] - 3*A[[1]], 2*A[[3]] - A[[1]]}
{{2, 3, -1}, {0, -13, 11}, {0, 7, 5}}
Now we choose "−13" as a pivot in the second row, so we multiply by 13 the third row \[ {\bf A}_3 = \begin{bmatrix} 2& 3& -1 \\ 0& -13& 11 \\ 0& 91& 65 \end{bmatrix} . \]
A3 = {A2[[1]], A2[[2]], 13*A2[[3]]}
{{2, 3, -1}, {0, -13, 11}, {0, 91, 65}}
Multiplying the second row by 7 and adding to the third row, we get
A4 = {A3[[1]], A3[[2]], A3[[3]] + 7*A3[[2]]}
{{2, 3, -1}, {0, -13, 11}, {0, 0, 142}}
\[ {\bf A}_4 = \begin{bmatrix} 2& 3& -1 \\ 0& -13& 11 \\ 0& 0& 142 \end{bmatrix} . \] This all-integer approach is easier for mental calculations.

Here is the general 3×3 case of this approach to finding the determinant. First, assuming 𝑎1,1 ≠ 0, we can rescale the lower rows.

\[ \mathbf{A} = \begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{2,1}a_{3,1} & a_{2,3} & a_{3,3} \end{bmatrix} \quad \Longrightarrow \quad \begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{1,1}a_{2,1} & a_{1,1}a_{2,2} & a_{1,1}a_{2,3} \\ a_{1,1}a_{3,1} & a_{1,1}a_{3,2} & a_{1,1}a_{3,3} \end{bmatrix} . \]
This rescales the determinant by 𝑎21,1. Now eliminate down the first column.
\[ \mathbf{A}_2 = \begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ 0 & a_{1,1}a_{2,2} - a_{1,2} a_{2,1} & a_{1,1} a_{2,3} - a_{2,1} a_{1,3} \\ 0 & a_{1,1}a_{3,2} - a_{3,1} a_{1,2} & a_{1,1}a_{3,3} - a_{3,1} a_{1,3} \end{bmatrix} . \]
Let C be the (1, 1)-minor. By Laplace the determinant of the above matrix is 𝑎1,1det(C). We thus have    
Example 1: Using Mathematica, we randonly generate a 4 × 4 matrix:
RandomInteger[{-9, 9}, {4, 4}]
{{0, 7, 2, 1}, {-6, -5, -8, 3}, {-3, 2, -7, 8}, {4, -1, 0, 8}}
However, the corresponding matrix \[ \begin{bmatrix} \fbox{0} & 7& 2& 1 \\ -6& -5& -8& 3 \\ -3& 2& -7& 8 \\ 4& -1& 0& 8 \end{bmatrix} . \] is not suitable for application of Chiò's method because the first pivot is zero. So we tray again, and generate the matrix
RandomInteger[{1, 9}, {4, 4}]
{{1, 2, 6, 3}, {3, 1, 7, 7}, {9, 3, 5, 5}, {7, 2, 6, 2}}
\[ \mathbf{A} = \begin{bmatrix} 1& 2& 6& 3 \\ 3& 1& 7& 7 \\ 9& 3& 5& 5 \\ 7& 2& 6& 2 \end{bmatrix} . \]    ■
End of Example 1
   
Example 1:    ■
End of Example 1
   
Example 1:    ■
End of Example 1

 

  1. Chiò, F., Mémoire sur les fonctions connues sous le noms De Résultantes Ou De Déterminans, Turin: E. Pons et C. Imprimeurs, 1853.
  2. C. L. Dodgson, Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetic Values, Proc. Roy. Soc. Ser. A, 15(1866), 150–155.
  3. H. Eves, An Introduction to the History of Mathematics, pages 405 and 493, Saunders College Publishing, 1990.
  4. H. Eves, Chio’s Expansion, §3.6 in Elementary Matrix Theory, New York: Dover, (1996), 129–136.
  5. ] J. R. Bunch and J. E. Hopcroft, Triangular factorization and inversion by fast matrix multiplication, Mathematics of Computation, 28 (1974), 231–236

 

  1. Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International