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# Assume $X, Y, Z, W$ and $P$ are matrices of order $2\times n, 3\times k, 2\times p, n\times 3$ and $p\times k$, respectively. The restriction on $n, p, k$ so that $PY + WY$ will be defined are:

\begin{array}{1 1} (A) \quad k = 3, p = n & (B) \quad k \text{ is arbitrary}, p = 2 \\ (C) \quad p \text{ is arbitrary}, k = 3 & (D) \quad k = 2, p = 3 \end{array}

Toolbox:
• Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix.
PY can be defined if the number of columns in $P=$ the number of rows in $Y$,
We know that the order of matrix P is $p\times k$ and the order of matrix Y is $3\times k.$ Therefore, for PY to be defined, $k$ must be equal to $3$ and the order of PY is $p\times k.$
WY can be defined if the number of columns in $W=$ the number of rows in $Y$,
We know that the order of matrix W is $n\times 3$ and the order of matrix Y is $3\times k$. The number of columns in $W =$ number of rows in $Y$ = 3. and the order of WY is $n\times k.$
Matrix $PY+WY$ is defined only when the $PY$ and $WY$ are of the same order.
Since the order of PY=$p\times k$ and the order of WY=$n\times k$, for $PY+WY$ to be define, $p$ must be equal to $n$.
Hence $PY+WY$ is defined when $k=3$,$p=n$
Thus correct option is (A).
edited Mar 1, 2013