Suppose $A$ is a $3\times 3$ symmetric matrix such that  :  $$\begin{pmatrix} a &amp; b &amp; 1   \end{pmatrix} A  \begin{pmatrix} a  \\  b\\ 1\end{pmatrix}  = ab -1$$ for all $a,b\in \mathbb{R}$  Question is to find no. of positive eigen values of $A$. and rank of $A$.  What i have observed so far is  :   As $A$ is symmetric matrix, it should have real eigen values. item As $x^TAx$ is not positive, for $x= \begin{pmatrix} a  \\  b\\ 1\end{pmatrix}$, ($a=b=\frac{1}{2}$) we see that not  all eigen values of $A$ are positive As $x^TAx$ is not negative, for $x= \begin{pmatrix} a  \\  b\\ 1\end{pmatrix}$, ($a=2,b=1$) we see that not  all eigen values of $A$ are Negative   So, $A$ does have positive eigen values and also negative eigen values.  Now, I have little less clarity particularly here :  If i take $a=b=1$ the, I have  $$\begin{pmatrix} 1 &amp; 1 &amp; 1   \end{pmatrix} A  \begin{pmatrix} 1  \\  1\\ 1\end{pmatrix}  = 1 -1=0$$  I want to conlcude from this that $$ A  \begin{pmatrix} 1  \\  1\\ 1\end{pmatrix}  =0\begin{pmatrix} 1  \\  1\\ 1\end{pmatrix}$$ somehow converting row marrix in one side to column matrix in other side.  If this is true, then i would have $0$ as one eigen value.  So, I have one positive, one negative, one $0$ eigen value.  So, Rank of $A$ should be $2$ and no. of positive eigen values is $1$.  Please help me to sort out some mistakes (if there are any) and help me to make this answer a little messier than what i have written.  This problem is already asked some time ago but, the answer for that was "to try with a  brute force" which i thought is not the only way to go.  I had some start up but that is not entire answer, so i can not post that as an answer there..   So, I thought i should ask here at the risk of getting negative votes.  P.S : Here is the link for a copy of this
Suppose A is a 3*3  symmetric matrix s.t.  $$\begin{pmatrix} x &amp; y &amp; 1 \\   \end{pmatrix} A  \begin{pmatrix} x  \\  y\\ 1\end{pmatrix}  = xy -1 $$  let p be the no. of positive eigen value of A and q = rank(A)-p, then p=? q=?  what matrix A should be I can't find??
Determine the number of ways to colour a 1 by n grid using the colours red and white if an even number of squares are to be coloured red. Is there a canonical explanation for how Ant-Man's weight works when he shrinks down that's consistent with the various events depicted in the film?    I think the explanation given in the film is roughly "the suit compresses the space in-between atoms, so your size decreases, your density increases, and your total mass stays the same", however it seems to be portrayed in contradictory ways.  For instance:   He can float in water, and even be sent shooting through the air by a relatively gentle flow of water filling a bathtub.  This wouldn't happen if he retained his original mass, as he'd essentially be the densest thing on the planet and should sink like a stone even in a vat of molten lead. Upon landing after being ejected from the bathtub, the ceramic tile underneath him shatters.  That takes a significant amount of force, and similar effects are used at other points in the movie where he falls onto (or straight through) various things. He's able to get sucked into a vacuum-cleaner.  One that didn't even appear to be a high-end, extra powerful vacuum-cleaner, let alone one capable of sucking up a 200 pound piece of dust.   He can run along the barrel of a gun as someone shoots it, apparently without causing them to drop the weapon or even have trouble continuing to aim it.     Does a more consistent explanation exist in other media, that would fit with the above examples?  Or is this simply a case of the power being made to do whatever happened to be most convenient at the time? patching SQL Server with Availability Group and Secondary Replica When does $\lim_{n\rightarrow \infty} a_n = 0$ imply $\sum_{n = 0}^{\infty} a_n$ converges? This is just a general question. I know the opposite statement is always true. I'm just asking when the converse holds. 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P.S.:  I had already partitioned my drives as mentioned above, and the D drive with 160GB was specifically left free for installing ubuntu. I would like to Select D drive during setup and install ubuntu here, while still keeping the already installed windows7 in C drive.  Thanks! Find the eigenvalues and eigenvectors with zeroes on the diagonal and ones everywhere else. I have been working on this problem for a couple hours and am completely stuck. Any help would be greatly appreciated.  Let $A$ be the $n \times n$ matrix which has zeros on the main diagonal and ones everywhere else. Find the eigenvalues and eigenvectors of $A$.