Identify statements `S_(1),S_(2),S_(3)` in order for true(T)false(F)
`S_(1)="If A"=[{:(,cos theta,-sin theta,0),(,sin theta,cos theta,0),(,0,0,1)]:}` then adjA=A'
`S_(2)="If A"=[{:(,a,0,0),(,0,b,0),(,0,0,c):}]"then "A^(-1)=[{:(,a,0,0),(,0,b,0),(,0,0,c):}]`
`S_(3)`: If B is non-singular matrix A is a squre matrix, then det `(B^(-1) AB)` =det(A)

Options

A. TTF
B. FTT
C. TFT
D. TTT

Answer

To determine the truth values of the statements \( S_1, S_2, \) and \( S_3 \), let's analyze each statement step by step.

### Step 1: Analyze Statement \( S_1 \)

**Statement:** If \( A = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \), then \( \text{adj} A = A' \).

1. **Find the Adjoint of \( A \):**
The adjoint of a matrix is the transpose of its cofactor matrix. For the given \( A \):
\[ A = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

The determinant of \( A \) can be calculated as:
\[ \text{det}(A) = \cos^2 \theta + \sin^2 \theta = 1 \]

The adjoint of \( A \) is given by:
\[ \text{adj} A = \text{det}(A) A^{-1} = A^{-1} \quad \text{(since det(A) = 1)} \]

The transpose \( A' \) (or \( A^T \)) is:
\[ A' = \begin{pmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

Since \( \text{adj} A \) is equal to \( A' \), we conclude that **Statement \( S_1 \) is True**.

### Step 2: Analyze Statement \( S_2 \)

**Statement:** If \( A = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \), then \( A^{-1} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \).

1. **Find the Inverse of \( A \):**
The inverse of a diagonal matrix \( A \) is given by:
\[ A^{-1} = \begin{pmatrix} \frac{1}{a} & 0 & 0 \\ 0 & \frac{1}{b} & 0 \\ 0 & 0 & \frac{1}{c} \end{pmatrix} \]

Since the given statement claims that \( A^{-1} = A \), which is not true unless \( a = b = c = 1 \). Therefore, **Statement \( S_2 \) is False**.

### Step 3: Analyze Statement \( S_3 \)

**Statement:** If \( B \) is a non-singular matrix and \( A \) is a square matrix, then \( \text{det}(B^{-1}AB) = \text{det}(A) \).

1. **Use the Property of Determinants:**
The property states:
\[ \text{det}(B^{-1}AB) = \text{det}(B^{-1}) \cdot \text{det}(A) \cdot \text{det}(B) \]
Since \( \text{det}(B^{-1}) = \frac{1}{\text{det}(B)} \), we have:
\[ \text{det}(B^{-1}AB) = \frac{1}{\text{det}(B)} \cdot \text{det}(A) \cdot \text{det}(B) = \text{det}(A) \]

Thus, **Statement \( S_3 \) is True**.

### Conclusion

- \( S_1 \): True
- \( S_2 \): False
- \( S_3 \): True

### Final Answer
The order of statements is:
- \( S_1 \) is True (T)
- \( S_2 \) is False (F)
- \( S_3 \) is True (T)