Assume X, Y, Z, W and P are matrices of order ` 2 xx n, 3 xx k, 2 xx p, n xx 3 andp xx k`, respectively. The restriction on n, k and p so that PY + WY will be difined are

Options

A. k = 3, p = n
B. k is arbitrary , p = 2
C. p is arbitrary , k = 3
D. k = 2, p = 3

Answer

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

### Step-by-Step Solution:

1. **Identify the dimensions of the matrices**:
- \( P \) is of order \( p \times k \)
- \( Y \) is of order \( 3 \times k \)
- \( W \) is of order \( n \times 3 \)

2. **Determine the condition for \( PY \)**:
- For the matrix multiplication \( PY \) to be defined, the number of columns in \( P \) must equal the number of rows in \( Y \).
- Thus, we have:
\[ k = 3 \]

3. **Determine the condition for \( WY \)**:
- For the matrix multiplication \( WY \) to be defined, the number of columns in \( W \) must equal the number of rows in \( Y \).
- Thus, we have:
\[ 3 = 3 \quad \text{(This condition is always satisfied)} \]

4. **Determine the condition for the sum \( PY + WY \)**:
- For \( PY + WY \) to be defined, both matrices must have the same dimensions.
- The dimensions of \( PY \) are \( p \times k \) and the dimensions of \( WY \) are \( n \times k \).
- Therefore, we need:
\[ p = n \]

5. **Summarize the restrictions**:
- From the analysis, we conclude that:
- \( k = 3 \)
- \( p = n \)

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