Important properties of Eigen values:

- Sum of all eigen values$=$Sum of leading diagonal(principle diagonal) elements=Trace of the matrix.
- Product of all Eigen values$=Det(A)= \mid A \mid$
- Any square diagonal(lower triangular or upper triangular) matrix eigen values are leading diagonal (principle diagonal)elements itself.

Example:$A=\begin{bmatrix} 1& 0& 0\\ 0&1 &0 \\ 0& 0& 1\end{bmatrix}$

`Diagonal matrix`

Eigenvalues are $1,1,1$

$B=\begin{bmatrix} 1& 9& 6\\ 0&1 &12 \\ 0& 0& 1\end{bmatrix}$

`Upper triangular matrix`

Eigenvalues are $1,1,1$

$C=\begin{bmatrix} 1& 0& 0\\ 8&1 &0 \\ 2& 3& 1\end{bmatrix}$

`Lower triangular matrix`

Eigenvalues are $1,1,1$

Now coming to the actual question

$R=\begin{bmatrix} 1 &2 &4 &8 \\ 1 &3 &9 &27 \\ 1 &4 &16 &64 \\ 1 &5 &25 &125 \end{bmatrix}$

$\mid R \mid =\begin{vmatrix} 1 &2 &4 &8 \\ 1 &3 &9 &27 \\ 1 &4 &16 &64 \\ 1 &5 &25 &125 \end{vmatrix}$

Perform

- $R4\rightarrow R_{4}-R_{3}$
- $R3\rightarrow R_{3}-R_{2}$
- $R2\rightarrow R_{2}-R_{1}$

$\implies \mid R \mid =\begin{vmatrix} 1 &2 &4 &8 \\ 0 &1 &5 &19 \\ 0 &1 &7 &37 \\ 0 &1 &9 &61 \end{vmatrix}$

Perform

- $R4\rightarrow R_{4}-R_{3}$
- $R3\rightarrow R_{3}-R_{2}$

$\implies \mid R \mid =\begin{vmatrix} 1 &2 &4 &8 \\ 0 &1 &5 &19 \\ 0 &0 &2 &18 \\ 0 &0 &2 &24 \end{vmatrix}$

Perform

- $R4\rightarrow R_{4}-R_{3}$

$\implies \mid R \mid =\begin{vmatrix} 1 &2 &4 &8 \\ 0 &1 &5 &19 \\ 0 &0 &2 &18 \\ 0 &0 &0 &6 \end{vmatrix}$

The `absolute value`

of product of Eigen values$=\text{Det}(A)= \text{Product of diagonal elements } =12.$