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STATEMENT -1 : A=(1)/(3){:[(1,-2,2),(-2,...

STATEMENT -1 : `A=(1)/(3){:[(1,-2,2),(-2,1,2),(-2,-2,-1)]:}` is an orthogonal matrix
and
STATEMENT-2 : If A and B are otthogonal, then AB is also orthogonal.

A

Statement -1 is True, Statement -2 is True , Statement -2 is a correct explanation for Statement-3

B

Statement-1 is True, Statement -2 is True , Statement -2 is NOT a correct explanation for Statement-3

C

Statement -1 is True, Statement -2 is False

D

Statement -1 is False , Statement -2 is True

Text Solution

AI Generated Solution

The correct Answer is:
To determine the validity of the statements regarding the matrix \( A \) and the properties of orthogonal matrices, we will analyze each statement step by step. ### Statement 1: Check if \( A \) is an orthogonal matrix 1. **Definition of Orthogonal Matrix**: A matrix \( A \) is orthogonal if \( A^T A = I \), where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix. 2. **Given Matrix**: \[ A = \frac{1}{3} \begin{pmatrix} 1 & -2 & 2 \\ -2 & 1 & 2 \\ -2 & -2 & -1 \end{pmatrix} \] 3. **Find the Transpose of \( A \)**: \[ A^T = \frac{1}{3} \begin{pmatrix} 1 & -2 & -2 \\ -2 & 1 & -2 \\ 2 & 2 & -1 \end{pmatrix} \] 4. **Compute \( A A^T \)**: \[ A A^T = \left( \frac{1}{3} \begin{pmatrix} 1 & -2 & 2 \\ -2 & 1 & 2 \\ -2 & -2 & -1 \end{pmatrix} \right) \left( \frac{1}{3} \begin{pmatrix} 1 & -2 & -2 \\ -2 & 1 & -2 \\ 2 & 2 & -1 \end{pmatrix} \right) \] Performing the matrix multiplication: \[ A A^T = \frac{1}{9} \begin{pmatrix} (1)(1) + (-2)(-2) + (2)(2) & (1)(-2) + (-2)(1) + (2)(2) & (1)(-2) + (-2)(-2) + (2)(-1) \\ (-2)(1) + (1)(-2) + (2)(2) & (-2)(-2) + (1)(1) + (2)(2) & (-2)(-2) + (1)(-2) + (2)(-1) \\ (-2)(1) + (-2)(-2) + (-1)(2) & (-2)(-2) + (-2)(1) + (-1)(2) & (-2)(-2) + (-2)(-2) + (-1)(-1) \end{pmatrix} \] After calculating each entry, we find: \[ A A^T = \frac{1}{9} \begin{pmatrix} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{pmatrix} = I \] 5. **Conclusion for Statement 1**: Since \( A A^T = I \), \( A \) is indeed an orthogonal matrix. Thus, Statement 1 is **true**. ### Statement 2: If \( A \) and \( B \) are orthogonal, then \( AB \) is also orthogonal. 1. **Orthogonal Matrix Properties**: If \( A \) and \( B \) are orthogonal, then \( A^T = A^{-1} \) and \( B^T = B^{-1} \). 2. **Check \( (AB)^T \)**: \[ (AB)^T = B^T A^T \] 3. **Check \( (AB)^{-1} \)**: \[ (AB)^{-1} = B^{-1} A^{-1} = B^T A^T \] 4. **Conclusion for Statement 2**: Since \( (AB)^T = (AB)^{-1} \), \( AB \) is also an orthogonal matrix. Thus, Statement 2 is **true**. ### Final Conclusion: Both statements are true, but they are independent of each other. Therefore, the answer is that both statements are correct. ---
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