Therefore, matrix $P Y$ will be defined if $k=3$

Consequently, $P Y$ will be of the order $p \times k$. Matrices $W$ and $Y$ are of the orders $n \times 3$ and $3 \times k$ respectively.

since the number of columns in $W$ is equal to the number of rows in $Y$, matrix $W Y$ is welldefined and is of the order $n\times k$.

Matrices $P Y$ and $W Y$ can be added only when their orders are the same.

However, $P Y$ is of the order $p \times k$ and $W Y$ is of the order $n \times k .$ Therefore. we must have

$p=n$

Thus, $k=3$ and $p=n$. are the restrictions on $n, \,k,$ and $p$ so that $P Y+W Y$ will be defined.