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Given the polynomial $f = (xy-yx)z \in \mathbb R[x,y,z]$, how can I evaluate it on matrices?

sage: A=random_matrix(QQ,2)
sage: A
[1 1]
[1 2]
sage: B=random_matrix(QQ,2)
sage: B
[ 0  0]
[-1  0]
sage: C=random_matrix(QQ,2)
sage: C
[ 1 -2]
[ 1 -1]
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: f = (x*y-y*x)*z
sage: f(A,B,C)
0
sage: (A*B-B*A)*C
[-1  2]
[ 0  1]

Given the polynomial $f = (xy-yx)z \in \mathbb R[x,y,z]$, Q[x,y,z]$, how can I evaluate it on matrices?

sage: A=random_matrix(QQ,2)
sage: A
[1 1]
[1 2]
sage: B=random_matrix(QQ,2)
sage: B
[ 0  0]
[-1  0]
sage: C=random_matrix(QQ,2)
sage: C
[ 1 -2]
[ 1 -1]
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: f = (x*y-y*x)*z
sage: f(A,B,C)
0
sage: (A*B-B*A)*C
[-1  2]
[ 0  1]

Given the polynomial $f = (xy-yx)z \in \mathbb Q[x,y,z]$, how How can I evaluate it on matrices?construct a map $M_2(\mathbb Q)^3 \to M_2(\mathbb Q), (A,B,C) \to (AB-BA)C$ where $M_2(\mathbb Q)$ denotes the space of 2x2 matrices.

sage: A=random_matrix(QQ,2)
sage: A
[1 1]
[1 2]
sage: B=random_matrix(QQ,2)
sage: B
[ 0  0]
[-1  0]
sage: C=random_matrix(QQ,2)
sage: C
[ 1 -2]
[ 1 -1]
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: f = (x*y-y*x)*z
sage: f(A,B,C)
0
sage: (A*B-B*A)*C
[-1  2]
[ 0  1]

How can I construct a map $M_2(\mathbb Q)^3 \to M_2(\mathbb Q), (A,B,C) \to (AB-BA)C$ where $M_2(\mathbb Q)$ denotes the space of 2x2 matrices.

sage: A=random_matrix(QQ,2)
sage: A
[1 1]
[1 2]
sage: B=random_matrix(QQ,2)
sage: B
[ 0  0]
[-1  0]
sage: C=random_matrix(QQ,2)
sage: C
[ 1 -2]
[ 1 -1]
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: f = (x*y-y*x)*z
sage: f(A,B,C)
0
sage: (A*B-B*A)*C
[-1  2]
[ 0  1]

How can I construct a map $M_2(\mathbb Q)^3 \to M_2(\mathbb Q), (A,B,C) \to \mapsto (AB-BA)C$ where $M_2(\mathbb Q)$ denotes the space of 2x2 matrices.

sage: A=random_matrix(QQ,2)
sage: A
[1 1]
[1 2]
sage: B=random_matrix(QQ,2)
sage: B
[ 0  0]
[-1  0]
sage: C=random_matrix(QQ,2)
sage: C
[ 1 -2]
[ 1 -1]
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: f = (x*y-y*x)*z
sage: f(A,B,C)
0
sage: (A*B-B*A)*C
[-1  2]
[ 0  1]

How can I construct a map $M_2(\mathbb Q)^3 \to M_2(\mathbb Q), (A,B,C) \mapsto (AB-BA)C$ where $M_2(\mathbb Q)$ denotes the space of 2x2 matrices.

sage: A=random_matrix(QQ,2)
x,y,z = var('x y z')
sage: A
[1 1]
[1 2]
f=(x*y-y*x)*z
sage: B=random_matrix(QQ,2)
sage: B
[ 0  0]
[-1  0]
sage: C=random_matrix(QQ,2)
sage: C
[ 1 -2]
[ 1 -1]
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: f = (x*y-y*x)*z
sage: f(A,B,C)
f
0
sage: (A*B-B*A)*C
[-1  2]
[ 0  1]

How can I construct a map $M_2(\mathbb Q)^3 \to M_2(\mathbb Q), (A,B,C) \mapsto (AB-BA)C$ where $M_2(\mathbb Q)$ denotes the space of 2x2 matrices.

sage: x,y,z = var('x y z')
sage: f=(x*y-y*x)*z
sage: f
0

How can I construct a map $M_2(\mathbb Q)^3 \to M_2(\mathbb Q), (A,B,C) \mapsto (AB-BA)C$ where $M_2(\mathbb Q)$ denotes the space of 2x2 matrices.

sage: x,y,z = var('x y z')
sage: f=(x*y-y*x)*z
sage: f
0