In the matrix equation $[A]\{X\}=\{R\}$,

[A] = 4 & 8 & 4\
4pt]
8 & 16 & -4\
4pt] 4 & -4 & 15 , \ = 2\
4pt]
1\
4pt] 4 , \ = 32\
4pt]
16\
4pt] 64 .

**One of the eigenvalues of the matrix $[A]$ is:**

To determine one of the eigenvalues of the given matrix $[\mathrm{A}]$, we will utilize the provided matrix equation $[\mathrm{A}]\{\mathrm{X}\}=\{\mathrm{R}\}$ and the definitions of eigenvalues and eigenvectors.

Matrix Equation Analysis

We are given the following matrices and vectors:

- The matrix $[\mathrm{A}]$ is: $$\mathrm{A}=\left[\begin{array}{ccc}4& 8& 4\\ 8& 16& -4\\ 4& -4& 15\end{array}\right]$$
- The vector $\{\mathrm{X}\}$ is: $$\mathrm{X}=\left\{\begin{array}{c}2\\ 1\\ 4\end{array}\right\}$$
- The vector $\{\mathrm{R}\}$ is: $$\mathrm{R}=\left\{\begin{array}{c}32\\ 16\\ 64\end{array}\right\}$$

The problem states the matrix equation $[\mathrm{A}]\{\mathrm{X}\}=\{\mathrm{R}\}$. Let's verify this by performing the matrix multiplication:

$$\mathrm{A}\mathrm{X}=\left[\begin{array}{ccc}4& 8& 4\\ 8& 16& -4\\ 4& -4& 15\end{array}\right]\left\{\begin{array}{c}2\\ 1\\ 4\end{array}\right\}$$

Calculating each component of the resulting vector:

- First component: $(4\times 2)+(8\times 1)+(4\times 4)=8+8+16=32$
- Second component: $(8\times 2)+(16\times 1)+(-4\times 4)=16+16-16=16$
- Third component: $(4\times 2)+(-4\times 1)+(15\times 4)=8-4+60=64$

So, the product $[\mathrm{A}]\{\mathrm{X}\}$ indeed results in the vector $\{\mathrm{R}\}$:

$$\mathrm{A}\mathrm{X}=\left\{\begin{array}{c}32\\ 16\\ 64\end{array}\right\}=\mathrm{R}$$

Eigenvalue Definition

An eigenvalue $\lambda$ and its corresponding eigenvector $\mathbf{v}$ for a square matrix $A$ satisfy the fundamental equation:

$$A\mathbf{v}=\lambda \mathbf{v}$$

In this equation, when matrix $A$ multiplies an eigenvector $\mathbf{v}$, the result is simply a scalar multiple ($\lambda$) of the same eigenvector $\mathbf{v}$.

Deriving the Matrix Eigenvalue

Now, let's compare the verified matrix equation $[\mathrm{A}]\{\mathrm{X}\}=\{\mathrm{R}\}$ with the eigenvalue definition $A\mathbf{v}=\lambda \mathbf{v}$.

We need to check if the vector $\{\mathrm{R}\}$ is a scalar multiple of the vector $\{\mathrm{X}\}$. Let's examine the components:

- Ratio of first components: $\frac{32}{2}=16$
- Ratio of second components: $\frac{16}{1}=16$
- Ratio of third components: $\frac{64}{4}=16$

Since all ratios are consistent, we can conclude that $\{\mathrm{R}\}=16\{\mathrm{X}\}$.

Substituting this relationship back into our matrix equation:

$$[\mathrm{A}]\{\mathrm{X}\}=16\{\mathrm{X}\}$$

This equation is precisely in the form of an eigenvalue equation, where $\{\mathrm{X}\}$ serves as an eigenvector and $16$ is the corresponding eigenvalue.

Final Eigenvalue Result

Based on our analysis, one of the eigenvalues of the matrix $[\mathrm{A}]$ is $16$.

Among the given options, $16$ is present.