Symmetric matrix with unit diagonal is invertible? Say I have a symmetric $M \times M$ matrix with unit diagonal. For example with $M=4$:
$$
\mathbf{A} = \left[\begin{array}{cccc}
1 & a  & b  & c  
\\
 a  & 1 & c  & d  
\\
 b  & c  & 1 & e  
\\
 c  & d  & e  & 1 
\end{array}\right]
$$
What way do you think is the easiest to show that this is an invertible matrix/has full rank? I have thought about showing that its determinant is always nonzero but the expression for the determinant becomes very intangible as $M$ is increased..
Oh and we can assume that the elements on the off-diagonal are not $1$ and they are positive. To make it clear, I actually have a matrix where $\mathbf{A}_{ij} = \exp(- \alpha \lVert c_i - c_j \rVert^2)$ where $\alpha$ is positive and all $c$s are different.
 A: Note: This is an incomplete answer

The general result you are looking for does not hold. For example, there exists a matrix for $M = 4$ with
$$
a = 3/10, \quad b = 2/10, \quad d = 9/10, \quad e = 4/10
$$
that fails to have full rank.
As I note in my comments, the approach laid out below only works for the matrix that you describe, i.e. the matrix with entries $\mathbf{A}_{ij} = \exp(- \alpha \lVert c_i - c_j \rVert^2)$. In fact, I'm having trouble generalizing this argument (since I'm not sure what necessary changes are being referred to here), so I'll stick to the case where $c_1,\dots,c_M \in \Bbb R$.
In order to show that a positive semidefinite matrix is positive definite, it suffices to show that for all non-zero vectors $x\in \Bbb R^M$, we have $x^TAx > 0$. Noting (as in this answer) that $\mathbf A_{ij} = E[e^{i(c_i - c_j)Z}]$, where $Z$ is a random variable with an $N(0,1)$ distribution. As the linked answer demonstrates, we have
$$
x^T \mathbf A x = \sum_{j,k=1}^M x_jx_k\mathbf A_{ij} = 
E\left[\left|\sum_{j=1}^M x_j e^{ic_j Z}\right|^2\right].
$$
Thus, it suffices to show that for $x \neq 0$, this expectation is non-zero. To this end, it suffices to show that if $x \neq 0$, then there exists at least one value of $t \in \Bbb R$ for which $\sum_{j=1}^M x_j e^{ic_j t} \neq 0$.
