I can never tell whether this color means the chicken needs another 30 seconds or a minute to finish or if it's properly cooked. I don't want to over-fry the chicken so that it doesn't end up dry. Is the color in the following image properly cooked or not?
Sometimes chicken breast can be thick and I never know how to tell when it's cooked! How can I be certain it is cooked all the way through?
I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and then the question.  True or False:   let $A,B \in \mathbb{Q}^{n \times n }$  Suppose $xI-A$ and $xI-B$ are equivalent then $\det(xI-A) = \det(xI-B)$.     I wrote in my notes this is true because any two matrices are equivalent in $\mathbb{Q[x]}^{n \times n}$ if and only if they have the same invariant factors.  Since the characteristic polynomial is a product of the invariant factors it follows that $\det(xI-A) = \det(xI-B)$.        True or False:   let $A,B \in \mathbb{Q}^{n \times n }$. If $\det(xI-A) = \det(xI-B)$ (in $\mathbb{Q}[x]$) then $xI-A$ and $xI-B$ are equivalent  in $\mathbb{Q}[x]^{n \times n}$   I think this is false because of the fact that equality of determinants is not strong enough to guarantee all invariant factors are equal but I do not have an example. How to avoid ArrayIndexOutOfBoundsException or IndexOutOfBoundsException? An awful identity We take place on $\mathbb C(x_1,...,x_r,x'_1,...,x'_p,u_0,...,u_r,u'_0,...,u'_p)$ with $r,p\in \mathbb N$     Show that :   $$\displaystyle{\sum_{i=1}^r \left(   \frac{\prod_{j=0}^r (u_j-x_i) \prod_{j=1}^p (x'_j-x_i) }{  \prod_{1\leq j\leq r, j\neq i     }(x_j-x_i) \prod_{j=0}^p (u'_j-x_i)}      \right)}-\displaystyle{\sum_{i=0}^p \left(   \frac{\prod_{j=0}^r (u'_i-u_j) \prod_{j=1}^p (u'_i-x'_j) }{  \prod_{0\leq j\leq p, j\neq i     }(u'_i-u'_j) \prod_{j=1}^r (u'_i-x_j)}      \right)}=\displaystyle{\sum_{i=1}^p x'_i +\sum_{i=0}^r u_i -\sum_{i=1}^r x_i-\sum_{i=0}^p u'_i}$$   I have seen this exercise in an old book at the library. Unfortunately there was no indication how can I solve the problem.   Anyway if someone can give me some ideas it will be greatful.   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