= Solution
The successive differences of the constant entries give a simple <LDL decomposition>:
$$
\boxed{L=\begin{pmatrix}1&0&0&0\\1&1&0&0\\1&1&1&0\\1&1&1&1\end{pmatrix},\qquad
D=\operatorname{diag}(1,4,9,\lambda-14),\qquad A=LDL^T}.
$$
For example, entry $(i,j)$ of $LDL^T$ is the sum of the first $\min(i,j)$ entries of $D$, directly verifying every entry of $A$. Because $L$ is an <invertible matrix>, $x^TAx=(L^Tx)^TD(L^Tx)$ is strictly positive for every nonzero <vector> exactly when every diagonal entry of $D$ is positive. Consequently
$$
\boxed{A\text{ is positive definite}\iff\lambda>14}.
$$
At $\lambda=14$ it is a <positive semidefinite matrix> and singular; for $\lambda<14$ its <quadratic form> takes both positive and negative values.
When $\lambda=30$, taking positive square roots of $D$ yields the <Cholesky decomposition>
$$
\boxed{A=CC^T,\qquad C=L\operatorname{diag}(1,2,3,4)=\begin{pmatrix}1&0&0&0\\1&2&0&0\\1&2&3&0\\1&2&3&4\end{pmatrix}}.
$$
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