Q.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. Then the restriction on $n$, $k$ and $p$ so that $PY+WY$ will be defined are:
(A) $k=3,\ p=n$
(B) $k$ is arbitrary, $p=2$
(C) $p$ is arbitrary, $k=3$
(D) $k=2,\ p=3$

The key idea is matrix multiplication compatibility: for two matrices to be multiplied, the number of columns in the first must equal the number of rows in the second. Addition is only defined if both matrices have the same order.

We are given orders:

$X_{2\times n},\ Y_{3\times k},\ Z_{2\times p},\ W_{n\times 3},\ P_{p\times k}$.

Step 1: For $PY$ to be defined, $P$ (order $p\times k$) times $Y$ (order $3\times k$) requires $k = 3$ (columns of $P$ = rows of $Y$). The product $PY$ then has order $p\times k = p\times 3$. …

The core idea here is matrix multiplication compatibility — the number of columns in the first matrix must equal the number of rows in the second. When adding two matrices, they must also have identical dimensions. So $PY+WY$ being defined means both $PY$ and $WY$ are individually defined, and their results are the same size.

Let’s unpack the given orders:

• $X$: $2 \times n$
• $Y$: $3 \times k$
• $Z$: $2 \times p$
• $W$: $n \times 3$
• $P$: $p \times k$

We only care about $P$, $W$, and $Y$.

1. Check $PY$: $P$ is $p \times k$, $Y$ is $3 \times k$. For multiplication $P \cdot Y$, the number of columns of $P$ must equal the number of rows of $Y$. That means $k = 3$. If $k \neq 3$, $PY$ is not defined at all, so the sum cannot exist. So $k = 3$ is necessary.

2. Check $WY$: $W$ is $n \times 3$, $Y$ is $3 \times k$. With $k = 3$ from step 1, $Y$ becomes $3 \times 3$. The columns of $W$ (3) match the rows of $Y$ (3), so $WY$ is defined for any $n$. Its resulting order is $n \times 3$.

3. Now $PY$ with $k=3$: $P$ is $p \times 3$, $Y$ is $3 \times 3$, so $PY$ is defined and its order is $p \times 3$. …

Method: Order restrictions for a sum of products

Use this when several matrices have symbolic orders and you must find the conditions that make an expression like $PY+WY$ defined.

Steps

Step 1: Make each product defined.

For every product in the expression, apply the inner-dimension rule (columns of the first $=$ rows of the second). Record each condition.

Step 2: Write down the order of each product.

The result of $X_{m\times n}Y_{n\times q}$ is $m\times q$ (the outer dimensions).

Step 3: Make the addition defined. …

Common Mistakes

Mistake 1: Checking only the multiplications and forgetting the addition condition.

Why it's wrong: even after $PY$ and $WY$ both exist, the sum needs them to have the same order; that requirement is what forces $p=n$. Correct approach: after finding each product's order ($p\times3$ and $n\times3$), equate them.

Mistake 2: Applying the inner-dimension rule to the wrong pair. …