Encoding messages with a matrix key

A pair of numbers \(\begin{pmatrix} x \\ y \end{pmatrix}\) is encoded as \(\mathbf{K}\begin{pmatrix} x \\ y \end{pmatrix}\), where \(\mathbf{K}\) is a \(2\times2\) key matrix with integer entries.

1. With \(\mathbf{K}=\begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\), a pair is encoded as \(\begin{pmatrix} 13 \\ 34 \end{pmatrix}\). Find the original pair.[3]
2. Explain, with an example, why \(\begin{pmatrix} 2 & 1 \\ 4 & 2 \end{pmatrix}\) cannot be used as a key.[2]
3. For the key \(\begin{pmatrix} k & 1 \\ 5 & 3 \end{pmatrix}\), where \(k\) is an integer, every encoded pair of integers must decode to a pair of integers. Find \(k\), justifying your answer.[3]