\title	{Propiedades de la multiplicación}
\author	{XIAO Gang}
\email	{xiao@unice.fr}
\format	{html}

\text{yes=True}
\text{no=False}
\text{bad=\yes,\no}
\integer{take=random(1,2)}
\text{AA=item(\take,A,B)}
\text{BB=item(\take,B,A)}
\text{AplusB=random(A + B, B + A)}
\text{AB=random(A,B)}
\text{ABBA=random(AB,BA)}
\integer{n=random(2..10)}

\matrix{data=
\(AB = 0),\(BA = 0),\no
\(AB \ne 0),\(BA \ne 0),\no
\(AB = 0),\(AB = BA),\no
\(AB = I_\n),\(BA = I_\n),\yes
\(AB \ne I_\n),\(BA \ne I_\n),\yes
\(AB = \AA),\(\BB = I_\n),\no
\(AB = 0) and \(\AA \ne 0),\(\BB = 0),\no
\(A \ne 0) and \(B \ne 0),\(AB \ne 0),\no
\(AB = BA),\((A+B)(A-B) = A^2 - B^2),\yes
\(AB \ne BA),\((A+B)(A-B) \ne A^2 - B^2),\yes
\((A+B)(A-B) = A^2 - B^2),\(AB = BA),\yes
\((A+B)(A-B) \ne A^2 - B^2),\(AB \ne BA),\yes
\((A+B)^2 = A^2 + 2AB + B^2),\(AB = BA),\yes
\(random(\AA(\AplusB) = \AA^2 + \AA\BB, (\AplusB)\AA = \AA^2 + \BB\AA)),\(AB = BA),\no
\(random(\AA(\AplusB) = \AA^2 + \BB\AA, (\AplusB)\AA = \AA^2 + \AA\BB)),\(AB = BA),\yes
\(AB = BA),\(random(\AA(\AplusB),(\AplusB)\AA) = \AA^2 + \ABBA),\yes
\(AB \ne BA),\(random(\AA(\AplusB) \ne \AA^2 + \BB\AA, (\AplusB)\AA \ne \AA^2 + \AA\BB)),\yes
\(AB \ne BA),\(random(\AA(\AplusB) \ne \AA^2 + \AA\BB, (\AplusB)\AA \ne \AA^2 + \BB\AA)),\no
\(random(AB = CB,BC = BA)),\(A = C),\no
\(A \ne C),\(random(AB \ne CB,BC \ne BA)),\no
\(random(A + B = C + B,B + C = B + A)),\(A = C),\yes
\(\BB = random(\AA^2, \AA+I_\n, \AA + \AA, \AA+\AA^2, \AA^2+I_\n)),\(AB = BA),\yes
\(\AplusB = random(0,I_\n)),\(AB = BA),\yes
\(\AplusB \ne random(0,I_\n)),\(AB \ne BA),\no
\(AB \ne BA),\(\AplusB \ne random(0,I_\n)),\yes
\(AB = BA),\(\AplusB = random(0,I_\n)),\no
\(AB \ne BA),\(A + B \ne B + A),\no
\(A + B = B + A),\(AB = BA),\no
\(\AA\BB = \AB),\(\BB\AA = \AB),\no
\(\AA\BB \ne \AB),\(\BB\AA \ne \AB),\no

}

text{data=
\(\AA\BB \ne \AB),\(\BB\AA \ne \AB),\no
}

\text{data=randomrow(\data)}
\text{hyp=item(1,\data)}
\text{cons=item(2,\data)}
\text{good=item(3,\data)}

\text{all=\hyp,\cons}
\if{C isin \all}{
 \text{mats=Sean \(A), \(B), \(C) tres matrices}
}{
 \if{B isin \all}{
  \text{mats=Sean \(A), \(B) dos matrices}
 }{
  \text{mats=Sea \(A) una matriz}
 }
}

\statement{\mats \n&times;\n. ¿Es cierta la siguiente afirmación?
<p><center>
 Si \hyp, entonces \cons.
</center><p>
\if{I_\n isin \hyp \cons}{
 (\(I_\n) es la matriz identidad \n&times;\n)
}
}

\choice{La respuesta}{\good}{\bad}