\title	{Operaciones algebraicas}
\author	{XIAO Gang}
\email	{xiao@unice.fr}
\format	{html}

\integer{n=random(4..9)}
\text{and={\rm\ and\ }}
\text{or={\rm\ or\ }}
\text{imply=\Rightarrow}
\text{pm=random(+,-)}
\text{det=det}
\text{tdet={\rm det}}
\text{inv=invertible}
\text{noninv=non-invertible}
\integer{a=random(2,3,4,5)}

\text{good=shuffle(
 \(\det(AB) = \det(BA)),
 \(\det(AB) = \det(A)\det(B)),
 \(\det(AB) = \det(B)\det(A)),
 \(\det(A^\a) = \det(A)^\a),
 \(\det(A^-1) = 1/\det(A)),
 \(A) \noninv \(=> \det(A) = 0),
 \(A) \inv \(\imply \tdet(A) \ne 0),
 \(\det(A) = 0 => A) \noninv,
 \(\tdet(A) \ne 0 \imply A) \inv,
 \(\det(\a*A) = \a^\n*\det(A)),
 \(A=I_\n => \det(A)=1),
 \(A=0 => \det(A)=0),
 \(A=B => \det(A)=\det(B)),
 \(\det(AB)=1 => A) \inv,
 \(\det(AB)=1 => B) \inv,
 \(\det(AB)=0 => \det(A)=0) or \(\det(B)=0)
)}
\text{bad=shuffle(
 \(\det(A \pm B) = \det(A) \pm \det(B)),
 \(\det(A^\a) = \a*\det(A)),
 \(\det(A+B) = \det(A)\det(B)),
 \(\det(AB) = \det(A) + \det(B)),
 \(\det(A^-1) = -\det(A)),
 \(\det(A+I_\n) = \det(A) + 1),
 \(\det(\a*A) = \a*\det(A)),
 \(\det(A) = 0 => A = 0),
 \(\det(A) = 1 => A = I_\n),
 \(\det(AB) = 1 => AB = I_\n),
 \(\det(AB) = \det(BA) => AB = BA),
 \(\det(A)=\det(B) => A=B),
 \(\det(AB)=1 => B=A^-1),
 \(\det(AB)=0 => A=0) or \(B=0),
 \(A \ne 0 \imply \tdet(A) \ne 0),
 \(A \ne I_\n \imply \tdet(A) \ne 1)
)}

\integer{badcnt=random(1,2)}
\integer{goodcnt=3-\badcnt}
\text{bads=item(1..\badcnt,1,1,1,1,1,1)}
\text{gb=item(1..3,\bads,0,0,0,0,0,0)}
\text{bads=item(1..\badcnt,\bad)}
\text{goods=item(1..3-\badcnt,\good)}
\text{order=shuffle(3)}
\text{list=item(\order,\bads,\goods)}
\text{gb=item(\order,\gb)}
\text{bad=wims(positionof item 1 in \gb)}
\text{good=wims(positionof item 0 in \gb)}
\text{l1=item(1,\list)}
\text{l2=item(2,\list)}
\text{l3=item(3,\list)}
\text{space=<td>&nbsp;&nbsp;&nbsp;}
\if{random(1,2) = 1}{
 \text{ask=no son siempre ciertas}
 \text{cond=falsas}
 \text{ans=\bad}
}{
 \text{ask=son siempre ciertas}
 \text{cond=correctas}
 \text{ans=\good}
}

\statement{Sean \(A,B) dos matrices \n&times;\n .
Entre las siguientes propiedades, ¿cuáles \ask?
<p><center><table border=0>
 <tr><td>1.\space<td>\l1
 <tr><td>2.\space<td>\l2
 <tr><td>3.\space<td>\l3
</table></center> <p>
}

\answer{Las propiedades \cond son}{\ans}{type=chset}