In matrix equation [A] {X} = {R}

One of the eigen values of Matrix [A] is

The correct option is 16.

To understand why this is the correct eigenvalue, let us recall the definition of an eigenvalue and its corresponding eigenvector. For any square matrix $\left[A\right]$, a non-zero vector $\left\{X\right\}$ is called an eigenvector if multiplying $\left[A\right]$ by $\left\{X\right\}$ yields a scalar multiple of $\left\{X\right\}$. This relationship is mathematically expressed by the characteristic equation:

$$\left[A\right]\left\{X\right\}=\lambda \left\{X\right\}$$
where $\lambda$ represents the eigenvalue corresponding to the eigenvector $\left\{X\right\}$.

From the first provided image, the matrix equation is defined as $\left[A\right]\left\{X\right\}=\left\{R\right\}$, where the specific matrices and vectors visible in the image are:

$$\left[A\right]=\left[\begin{array}{ccc}4& 8& 4\\ 8& 16& -4\\ 4& -4& 15\end{array}\right],\left\{X\right\}=\left[\begin{array}{c}2\\ 1\\ 4\end{array}\right],\left\{R\right\}=\left[\begin{array}{c}32\\ 16\\ 64\end{array}\right]$$

Let us verify the matrix multiplication of $\left[A\right]$ and the vector $\left\{X\right\}$ step-by-step:

$$\left[A\right]\left\{X\right\}=\left[\begin{array}{ccc}4& 8& 4\\ 8& 16& -4\\ 4& -4& 15\end{array}\right]\left[\begin{array}{c}2\\ 1\\ 4\end{array}\right]$$
Calculating each component of the resulting vector:
• First element: $(4\times 2)+(8\times 1)+(4\times 4)=8+8+16=32$
• Second element: $(8\times 2)+(16\times 1)+(-4\times 4)=16+16-16=16$
• Third element: $(4\times 2)+(-4\times 1)+(15\times 4)=8-4+60=64$

Thus, the resulting product vector $\left\{R\right\}$ is:

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

Now, we can factor out a common scalar value from the vector $\left\{R\right\}$ to relate it back to the original eigenvector $\left\{X\right\}$:

$$\left[\begin{array}{c}32\\ 16\\ 64\end{array}\right]=16\times \left[\begin{array}{c}2\\ 1\\ 4\end{array}\right]$$
This perfectly matches the eigenvalue equation:

$$\left[A\right]\left\{X\right\}=\lambda \left\{X\right\}$$
where $\lambda =16$ and $\left\{X\right\}={\left[214\right]}^T$ is the corresponding eigenvector.
Therefore, $16$ is one of the eigenvalues of the matrix $\left[A\right]$.