Question
If $A = \begin{bmatrix} \sin 2\theta & 2 \sin^2 \theta - 1 & 0 \\ \cos 2\theta & 2 \sin \theta \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then which of the following statements is/are correct?
1. $A^{-1} = \text{adj}A$
2. $A$ is skew-symmetric matrix
3. $A^{-1} = A^T$
Select the correct answer using the code given below :
This question was previously asked in
NDA 2 2024 GAT Question Paper (01-Sep-2024)
The correct answer is
1 and 3
To solve this question, we will analyze the given matrix \( A \) and evaluate the statements provided in the options.
The matrix \( A \) is:
| \(\sin 2\theta\) | \(2 \sin^2 \theta - 1\) | 0 |
| \(\cos 2\theta\) | \(2 \sin \theta \cos \theta\) | 0 |
| 0 | 0 | 1 |
Let's evaluate each statement:
- **Statement 1:** \( A^{-1} = \text{adj}A \)
- According to matrix properties, if \(\det(A) = 1\), then \( A^{-1} = \text{adj}(A) \).
- The determinant of \( A \), a 3x3 matrix with zero elements making it simple to calculate, is:
- Hence, \(\det(A) = 1\), so this statement is correct.
- **Statement 2:** \( A \) is a skew-symmetric matrix.
- A skew-symmetric matrix is defined as a matrix satisfying \( A = -A^T \).
- Calculating \( A^T \):
- Clearly, \( A \neq -A^T \) as the matrix isn’t equal to its negative transpose.
- Thus, \( A \) is not skew-symmetric, making this statement incorrect.
- **Statement 3:** \( A^{-1} = A^T \).
- Given that \(\det(A) = 1\), if \( A^T = \text{adj}(A) \), then \( A^{-1} = A^T \).
- Evaluating to ensure \( A^T \) turns out to be equal to \(\text{adj}(A)\), we find the entries match.
- Therefore, this statement is confirmed accurate due to the property of orthogonal matrices wherein \(\det = 1\), leading to equality of inverses and transposes.
In conclusion, the correct statements regarding the matrix \( A \) are options 1 and 3.
Therefore, the correct answer is 1 and 3.
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