Let P be the set of all non - singular matrices of order 3 over R and Q be the set of all orthogonal matrices of order 3 over R. Then

Options

A. P is proper subset of Q
B. Neither P is proper subset of Q nor Q is proper subset of P
C. `PnnQ=phi` the void set
D. Q is proper subset of P

Answer

To solve the problem, we need to analyze the sets \( P \) and \( Q \) defined in the question.

### Step 1: Define the sets \( P \) and \( Q \)

- **Set \( P \)**: This is the set of all non-singular matrices of order 3 over \( \mathbb{R} \). A matrix is non-singular if its determinant is non-zero.
- **Set \( Q \)**: This is the set of all orthogonal matrices of order 3 over \( \mathbb{R} \). A matrix \( Q \) is orthogonal if it satisfies the condition \( Q Q^T = I \), where \( I \) is the identity matrix. For orthogonal matrices, the determinant is either \( 1 \) or \( -1 \).

### Step 2: Analyze the properties of the determinants

- For any matrix \( A \) in set \( P \), the determinant \( \det(A) \) can be any non-zero real number. Therefore, \( \det(P) \neq 0 \).
- For any matrix \( B \) in set \( Q \), the determinant \( \det(B) \) must be either \( 1 \) or \( -1 \). Thus, \( \det(Q) = \{1, -1\} \).

### Step 3: Establish the relationship between sets \( P \) and \( Q \)

Since all orthogonal matrices are non-singular (because their determinants are non-zero), we can conclude that every orthogonal matrix is also a non-singular matrix. Therefore, every matrix in set \( Q \) is also in set \( P \).

### Step 4: Determine if \( Q \) is a proper subset of \( P \)

- Since \( Q \) contains matrices whose determinants are specifically \( 1 \) or \( -1 \), and \( P \) contains all non-zero determinants (which includes values other than \( 1 \) and \( -1 \)), it follows that \( Q \) is a proper subset of \( P \).
- This means that while all elements of \( Q \) are in \( P \), there are elements in \( P \) that are not in \( Q \) (for example, matrices with determinants like \( 2 \), \( -3 \), etc.).

### Conclusion

Thus, we conclude that \( Q \) is a proper subset of \( P \).

### Final Answer

\( Q \subset P \) (Q is a proper subset of P).

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