Why is the above matrix singular? Let
$$A=\begin{bmatrix}1&2&3\\3&4&5\\5&6&7\end{bmatrix}$$
with $\det(A) = 0$. We know if $\det(A) = 0$, then $A$ is singular. Also, we know if $A$ is singular, columns and rows are linearly dependent. Why are the columns or rows linearly dependent?

We see that in every row the elements increase by 1 when going from column 1 to column 2 to column 3. In other words:
$$
\begin{pmatrix}2\\4\\6\end{pmatrix}-\begin{pmatrix}1\\3\\5\end{pmatrix}=
\begin{pmatrix}1\\1\\1\end{pmatrix}=
\begin{pmatrix}3\\5\\7\end{pmatrix}-\begin{pmatrix}2\\4\\6\end{pmatrix}
$$
Rearranging, you will get:
$$
2\begin{pmatrix}2\\4\\6\end{pmatrix}=
\begin{pmatrix}1\\3\\5\end{pmatrix}+\begin{pmatrix}3\\5\\7\end{pmatrix}
$$
which is clearly a linear relation between the columns. Thus, columns are not linearly independent
