2015-11-19-120447.sagews

f(x,y,z) = (y*z,x*z,x*y)

var('a1 a2 a3 a4 a5 a6 a7 a8 a9')
A = Matrix([[a1,a2,a3],[a4,a5,a6],[a7,a8,a9]]); show(A)

(a1, a2, a3, a4, a5, a6, a7, a8, a9)

$\displaystyle \left(\begin{array}{rrr} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{array}\right)$

var('b1 b2 b3 b4 b5 b6 b7 b8 b9')
B = Matrix([[b1,b2,b3],[b4,b5,b6],[b7,b8,b9]]); show(B)

(b1, b2, b3, b4, b5, b6, b7, b8, b9)

$\displaystyle \left(\begin{array}{rrr} b_{1} & b_{2} & b_{3} \\ b_{4} & b_{5} & b_{6} \\ b_{7} & b_{8} & b_{9} \end{array}\right)$

var('x11 x12 y11 y12')
X1 = Matrix([[x11],[x12], [1]])
Y1 = Matrix([[y11],[y12], [1]])

(x11, x12, y11, y12)

(A*X1)[2]

(a7*x11 + a8*x12 + a9)

E = B*f((A*X1)[0][0],(A*X1)[1][0],(A*X1)[2][0])
show(E[0].simplify_full())
show(E[1].simplify_full())
show(E[2].simplify_full())

$\displaystyle a_{6} a_{9} b_{1} + a_{3} a_{9} b_{2} + a_{3} a_{6} b_{3} + {\left(a_{4} a_{7} b_{1} + a_{1} a_{7} b_{2} + a_{1} a_{4} b_{3}\right)} x_{11}^{2} + {\left(a_{5} a_{8} b_{1} + a_{2} a_{8} b_{2} + a_{2} a_{5} b_{3}\right)} x_{12}^{2} + {\left({\left(a_{6} a_{7} + a_{4} a_{9}\right)} b_{1} + {\left(a_{3} a_{7} + a_{1} a_{9}\right)} b_{2} + {\left(a_{3} a_{4} + a_{1} a_{6}\right)} b_{3}\right)} x_{11} + {\left({\left(a_{6} a_{8} + a_{5} a_{9}\right)} b_{1} + {\left(a_{3} a_{8} + a_{2} a_{9}\right)} b_{2} + {\left(a_{3} a_{5} + a_{2} a_{6}\right)} b_{3} + {\left({\left(a_{5} a_{7} + a_{4} a_{8}\right)} b_{1} + {\left(a_{2} a_{7} + a_{1} a_{8}\right)} b_{2} + {\left(a_{2} a_{4} + a_{1} a_{5}\right)} b_{3}\right)} x_{11}\right)} x_{12}$

$\displaystyle a_{6} a_{9} b_{4} + a_{3} a_{9} b_{5} + a_{3} a_{6} b_{6} + {\left(a_{4} a_{7} b_{4} + a_{1} a_{7} b_{5} + a_{1} a_{4} b_{6}\right)} x_{11}^{2} + {\left(a_{5} a_{8} b_{4} + a_{2} a_{8} b_{5} + a_{2} a_{5} b_{6}\right)} x_{12}^{2} + {\left({\left(a_{6} a_{7} + a_{4} a_{9}\right)} b_{4} + {\left(a_{3} a_{7} + a_{1} a_{9}\right)} b_{5} + {\left(a_{3} a_{4} + a_{1} a_{6}\right)} b_{6}\right)} x_{11} + {\left({\left(a_{6} a_{8} + a_{5} a_{9}\right)} b_{4} + {\left(a_{3} a_{8} + a_{2} a_{9}\right)} b_{5} + {\left(a_{3} a_{5} + a_{2} a_{6}\right)} b_{6} + {\left({\left(a_{5} a_{7} + a_{4} a_{8}\right)} b_{4} + {\left(a_{2} a_{7} + a_{1} a_{8}\right)} b_{5} + {\left(a_{2} a_{4} + a_{1} a_{5}\right)} b_{6}\right)} x_{11}\right)} x_{12}$

$\displaystyle a_{6} a_{9} b_{7} + a_{3} a_{9} b_{8} + a_{3} a_{6} b_{9} + {\left(a_{4} a_{7} b_{7} + a_{1} a_{7} b_{8} + a_{1} a_{4} b_{9}\right)} x_{11}^{2} + {\left(a_{5} a_{8} b_{7} + a_{2} a_{8} b_{8} + a_{2} a_{5} b_{9}\right)} x_{12}^{2} + {\left({\left(a_{6} a_{7} + a_{4} a_{9}\right)} b_{7} + {\left(a_{3} a_{7} + a_{1} a_{9}\right)} b_{8} + {\left(a_{3} a_{4} + a_{1} a_{6}\right)} b_{9}\right)} x_{11} + {\left({\left(a_{6} a_{8} + a_{5} a_{9}\right)} b_{7} + {\left(a_{3} a_{8} + a_{2} a_{9}\right)} b_{8} + {\left(a_{3} a_{5} + a_{2} a_{6}\right)} b_{9} + {\left({\left(a_{5} a_{7} + a_{4} a_{8}\right)} b_{7} + {\left(a_{2} a_{7} + a_{1} a_{8}\right)} b_{8} + {\left(a_{2} a_{4} + a_{1} a_{5}\right)} b_{9}\right)} x_{11}\right)} x_{12}$

var('k')
k*Y1

[k*y11] [k*y12] [ k]