1)

there is $\displaystyle A=\left(\begin{array}{ccc}4 & -2 & 4\\-2 & 1 & -2\\4 & -2 & 4\end{array}\right)$

A.we define $\displaystyle f:R^{3}xR^{3}->R$ by $\displaystyle f(x,y)=(Ax,y)$ for every $\displaystyle y=(y_{1},y_{2},y_{3})^{t}$and $\displaystyle x=(x_{1},x_{2},x_{3})^{t}$ in $\displaystyle R^{3}$

does f define inner product in $\displaystyle R^{3}$?

how i tried:

in the book by definition inner product is following these rules:

(u,v)=(v,u)

(u+v,w)=(u,w)+(v,w)

(ku,v)=k(u,v)

$\displaystyle (v,v)\geq0$

i dont know how to apply these rules here

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B.find a simetric matrices B so $\displaystyle B^{2}=A $

(clue: use cayley-Hamilton law)

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how i tried:

hamilton law say that the representative matrice turns to zero the minimal polinomial M(A)=0

any guidence

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