the restriction on n, k and p so that PY +Wywill be defined are :

Options

A. K=3,p=n
B. K is arbitrary , p=2
C. p is arobitray , k-3
D. `k=2,p=3`

Explanation

To determine the restrictions on \( n \), \( k \), and \( p \) such that \( PY + WY \) is defined, we need to analyze the orders of the matrices involved in the expression.

### Step-by-Step Solution:

1. **Identify the Orders of the Matrices:**
- Let the orders of the matrices be defined as follows:
- \( P \) is of order \( p \times k \)
- \( Y \) is of order \( 3 \times k \)
- \( W \) is of order \( n \times 3 \)

2. **Determine the Order of \( PY \):**
- The multiplication \( PY \) is defined if the number of columns in \( P \) is equal to the number of rows in \( Y \).
- Therefore, for \( P \) of order \( p \times k \) and \( Y \) of order \( 3 \times k \):
- The condition for multiplication is \( k = 3 \).
- The resulting order of \( PY \) will be \( p \times k \) or \( p \times 3 \).

3. **Determine the Order of \( WY \):**
- The multiplication \( WY \) is defined if the number of columns in \( W \) is equal to the number of rows in \( Y \).
- Therefore, for \( W \) of order \( n \times 3 \) and \( Y \) of order \( 3 \times k \):
- The condition for multiplication is \( 3 = 3 \) (which is always true).
- The resulting order of \( WY \) will be \( n \times k \) or \( n \times 3 \).

4. **Condition for Addition \( PY + WY \):**
- For the addition \( PY + WY \) to be defined, both matrices must have the same order.
- From the previous steps, we have:
- Order of \( PY \) is \( p \times 3 \)
- Order of \( WY \) is \( n \times 3 \)
- Therefore, we need:
- \( p = n \)

5. **Summary of Conditions:**
- From the analysis, we have two conditions:
- \( k = 3 \)
- \( p = n \)

### Final Answer:
The restrictions on \( n \), \( k \), and \( p \) so that \( PY + WY \) will be defined are:
- \( k = 3 \)
- \( p = n \)