Untitled -- Sage

var('p,q,v1,v2,v3,v4');

(p, q, v1, v2, v3, v4)

(p, q, v1, v2, v3, v4)

A=Matrix(SR,[[p*q,q,p,1,v1],[q,p*q,1,p,v2],[p,1,p*q,q,v3],[1,1,1,1,v4]]);A

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr} p q & q & p & 1 & v_{1} \\ q & p q & 1 & p & v_{2} \\ p & 1 & p q & q & v_{3} \\ 1 & 1 & 1 & 1 & v_{4} \end{array}\right)$

A.rref()

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & \frac{{\left(\frac{q - \frac{1}{q}}{{\left(p q - \frac{p}{q}\right)} q} + \frac{p - \frac{1}{p}}{{\left(p q - \frac{q}{p}\right)} p} - \frac{1}{p q}\right)} {\left(\frac{{\left(\frac{1}{q} - 1\right)} {\left(v_{3} - \frac{v_{1}}{q}\right)}}{p q - \frac{p}{q}} + \frac{{\left(\frac{1}{p} - 1\right)} {\left(v_{2} - \frac{v_{1}}{p}\right)}}{p q - \frac{q}{p}} + v_{4} - \frac{v_{1}}{p q}\right)}}{\frac{{\left(\frac{1}{q} - 1\right)} {\left(q - \frac{1}{q}\right)}}{p q - \frac{p}{q}} + \frac{{\left(\frac{1}{p} - 1\right)} {\left(p - \frac{1}{p}\right)}}{p q - \frac{q}{p}} - \frac{1}{p q} + 1} - \frac{v_{3} - \frac{v_{1}}{q}}{{\left(p q - \frac{p}{q}\right)} q} - \frac{v_{2} - \frac{v_{1}}{p}}{{\left(p q - \frac{q}{p}\right)} p} + \frac{v_{1}}{p q} \\ 0 & 1 & 0 & 0 & \frac{v_{2} - \frac{v_{1}}{p}}{p q - \frac{q}{p}} - \frac{{\left(p - \frac{1}{p}\right)} {\left(\frac{{\left(\frac{1}{q} - 1\right)} {\left(v_{3} - \frac{v_{1}}{q}\right)}}{p q - \frac{p}{q}} + \frac{{\left(\frac{1}{p} - 1\right)} {\left(v_{2} - \frac{v_{1}}{p}\right)}}{p q - \frac{q}{p}} + v_{4} - \frac{v_{1}}{p q}\right)}}{{\left(p q - \frac{q}{p}\right)} {\left(\frac{{\left(\frac{1}{q} - 1\right)} {\left(q - \frac{1}{q}\right)}}{p q - \frac{p}{q}} + \frac{{\left(\frac{1}{p} - 1\right)} {\left(p - \frac{1}{p}\right)}}{p q - \frac{q}{p}} - \frac{1}{p q} + 1\right)}} \\ 0 & 0 & 1 & 0 & \frac{v_{3} - \frac{v_{1}}{q}}{p q - \frac{p}{q}} - \frac{{\left(q - \frac{1}{q}\right)} {\left(\frac{{\left(\frac{1}{q} - 1\right)} {\left(v_{3} - \frac{v_{1}}{q}\right)}}{p q - \frac{p}{q}} + \frac{{\left(\frac{1}{p} - 1\right)} {\left(v_{2} - \frac{v_{1}}{p}\right)}}{p q - \frac{q}{p}} + v_{4} - \frac{v_{1}}{p q}\right)}}{{\left(p q - \frac{p}{q}\right)} {\left(\frac{{\left(\frac{1}{q} - 1\right)} {\left(q - \frac{1}{q}\right)}}{p q - \frac{p}{q}} + \frac{{\left(\frac{1}{p} - 1\right)} {\left(p - \frac{1}{p}\right)}}{p q - \frac{q}{p}} - \frac{1}{p q} + 1\right)}} \\ 0 & 0 & 0 & 1 & \frac{\frac{{\left(\frac{1}{q} - 1\right)} {\left(v_{3} - \frac{v_{1}}{q}\right)}}{p q - \frac{p}{q}} + \frac{{\left(\frac{1}{p} - 1\right)} {\left(v_{2} - \frac{v_{1}}{p}\right)}}{p q - \frac{q}{p}} + v_{4} - \frac{v_{1}}{p q}}{\frac{{\left(\frac{1}{q} - 1\right)} {\left(q - \frac{1}{q}\right)}}{p q - \frac{p}{q}} + \frac{{\left(\frac{1}{p} - 1\right)} {\left(p - \frac{1}{p}\right)}}{p q - \frac{q}{p}} - \frac{1}{p q} + 1} \end{array}\right)$

A.swap_rows(0,3);

A;

[  1   p   q p*q  v4]
[  q p*q   1   p  v2]
[  p   1 p*q   q  v3]
[p*q   q   p   1  v1]

[  1   p   q p*q  v4]
[  q p*q   1   p  v2]
[  p   1 p*q   q  v3]
[p*q   q   p   1  v1]

A.add_multiple_of_column(1,0,-p);A

[           1            0            q          p*q           v4]
[           0            0     -q^2 + 1   -p*q^2 + p   -q*v4 + v2]
[           0     -p^2 + 1            0   -p^2*q + q   -p*v4 + v3]
[           0   -p^2*q + q   -p*q^2 + p -p^2*q^2 + 1 -p*q*v4 + v1]

[           1            0            q          p*q           v4]
[           0            0     -q^2 + 1   -p*q^2 + p   -q*v4 + v2]
[           0     -p^2 + 1            0   -p^2*q + q   -p*v4 + v3]
[           0   -p^2*q + q   -p*q^2 + p -p^2*q^2 + 1 -p*q*v4 + v1]

Untitled

3076 days ago by jimlb