Answer
See below
Work Step by Step
Given $$\begin{array}{c}{A=\left[\begin{array}{ccc}
{1} & {2} & 3& 4 & 5\\
{5} & {4} & 3 & 2 & 1 \\
2 & 3 & 4 & 5 & 6\\
6& 5 & 4 & 3 & 2\\
0 & 2 & 4 & 6 & 8 \end{array}\right] }\end{array} $$
So, we get
$$A=\begin{vmatrix} {1} & {2} & 3& 4 & 5\\
{5} & {4} & 3 & 2 & 1 \\
2 & 3 & 4 & 5 & 6\\
6& 5 & 4 & 3 & 2\\
0 & 2 & 4 & 6 & 8
\end{vmatrix} \rightarrow \begin{vmatrix}
{1} & {2} & 3& 4 & 5\\
{0} & {-6} & -12 & -18 & -24 \\
0 & -1 & -2 & -3 & -4\\
0 & -7 & -14 & -21 & -28\\
0 & 2 & 4 & 6 & 8
\end{vmatrix} \rightarrow \begin{vmatrix}
{1} & {2} & 3& 4 & 5\\
{0} & {1} & 2 & 3 & 4 \\
0 & -1 & -2 & -3 & -4\\
0 & -7 & -14 & -21 & -28\\
0 & 2 & 4 & 6 & 8
\end{vmatrix} \rightarrow -6\begin{vmatrix}
{1} & {2} & 3& 4 & 5\\
{0} & {1} & 2 & 3 & 4 \\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0
\end{vmatrix}$$
\begin{array}{l}{\text { Determinant of triangular matrix is the product of its diagonal elements, so: }} \\ {\text {det }A=-6 \cdot 1 \cdot 1 \cdot 0 \cdot 0 \cdot 0=0}\end{array}