Show a matrix is positive definite I was looking at UCLA's Department of Mathematics's Fall 2015 Basic Qualifying Exam, and question #12 is the following.


  Show that the following matrix is positive definite.
  
  $$M=\begin{pmatrix}
2 &amp; 1 &amp; 1 &amp; \ldots &amp; 1\\
1 &amp; 3 &amp; 1 &amp; \ldots &amp; 1\\
1 &amp; 1 &amp; 4 &amp; \ldots &amp; 1\\
\vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
1 &amp; 1&amp; 1&amp; \ldots &amp;n+1
\end{pmatrix}$$


I wasn't super sure how one should proceed; showing $x^TMx&gt;0$ for all $x \neq 0$ seems like a pain, and I was curious if there was a slick way to do this. Other ways would be to write $M = AA^T$ or go through the characteristic polynomial, but I'm not super sure about those either. If anyone can give some hints, that would be great!
Write 
$$M=\begin{pmatrix}
1 &amp; 0 &amp; 0 &amp; \ldots &amp; 0\\
0 &amp; 2 &amp; 0 &amp; \ldots &amp; 0\\
0 &amp; 0 &amp; 3 &amp; \ldots &amp; 0\\
\vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
0 &amp; 0&amp; 0&amp; \ldots &amp;n
\end{pmatrix}+
\begin{pmatrix}
1 &amp; 1 &amp; 1 &amp; \ldots &amp; 1\\
1 &amp; 1 &amp; 1 &amp; \ldots &amp; 1\\
1 &amp; 1 &amp; 1 &amp; \ldots &amp; 1\\
\vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
1 &amp; 1&amp; 1&amp; \ldots &amp;1
\end{pmatrix}.$$

Now it's easy to see that the first of these matrices is positive-definite, while the second is positive-semidefinite (in particular, its eigenvalues are $n$ and $0$).
Maybe you can consult https://en.wikipedia.org/wiki/Sylvester%27s_criterion. Proceed by induction.