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Assume $X,\, Y,\, Z, W$ and $P$ are the matrices of order $2 \times n, \,3 \times k,\, 2 \times p, \,n \times 3$ and $p \times k$ respectively. The restriction on $n,\, k$ and $p$ so that $P Y+W Y$ will be defined are :
$p$ is arbitrary, $k=3$
$k$ is arbitrary, $p=2$
$k=3$, $p=n$
$k=2$, $p=3$
Solution
Matrices $P$ and $Y$ are of the orders $p \times k$ and $3 \times k$ respectively.
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.
