Determinant Bound for a Matrix Squeezed between a Negative Multiple of the Identity and Zero

Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, with the absolute value written $|\cdot|$. Let $\lambda\in\mathbb{R}$ with $0\le\lambda$, and let $A$ be a symmetric real $n\times n$ matrix.

Let $I_{n}$ be the identity matrix, let $0_{n}$ be the real $n\times n$ matrix all of whose entries are $0$, let $\mu M$ denote the scalar multiple of a matrix $M$ by $\mu\in\mathbb{R}$, and let $\preceq$ be the positive semidefinite ordering; the matrices $0_{n}$ and $(-\lambda)I_{n}$ are symmetric, since their entries satisfy $M_{ij}=M_{ji}$. Determinants are read as in Row Properties of the Determinant and powers are those of Natural Number Power of an Element of a Field.

If

then