Let P and Q are two non-singular square matrices of order `3xx3` such that `PQ = P and QP =Q` . If det. `(P+Q) ne0` then the value of det. ``(P+Q) +` det (P-Q) ` is___________.

Answer

To solve the problem, we need to find the value of \( \det(P + Q) + \det(P - Q) \) given the conditions \( PQ = P \) and \( QP = Q \) for two non-singular \( 3 \times 3 \) matrices \( P \) and \( Q \).

### Step-by-step Solution:

1. **Understanding the Conditions**:
- From \( PQ = P \), we can rearrange it to \( P - PQ = 0 \) or \( P(I - Q) = 0 \).
- Since \( P \) is non-singular, this implies \( I - Q = 0 \) or \( Q = I \).
- Similarly, from \( QP = Q \), we can rearrange it to \( Q - QP = 0 \) or \( Q(I - P) = 0 \).
- Since \( Q \) is non-singular, this implies \( I - P = 0 \) or \( P = I \).

2. **Finding \( P + Q \) and \( P - Q \)**:
- If both \( P \) and \( Q \) are equal to the identity matrix \( I \), we have:
\[ P + Q = I + I = 2I \]
\[ P - Q = I - I = 0 \]

3. **Calculating Determinants**:
- The determinant of \( P + Q \):
\[ \det(P + Q) = \det(2I) = 2^3 \cdot \det(I) = 8 \]
- The determinant of \( P - Q \):
\[ \det(P - Q) = \det(0) = 0 \]

4. **Final Calculation**:
- Now, we can find the required value:
\[ \det(P + Q) + \det(P - Q) = 8 + 0 = 8 \]

Thus, the value of \( \det(P + Q) + \det(P - Q) \) is **8**.