A `3xx3` matrix A, with 1st row elements as 2,-1,-1 respectively, is modified as below to get another matrix B.
`R_1` elements of A go to `R_3` of matrix C
`R_2` elements of A go to `R_1` of matrix C
`R_2` elements of A to `R_1` of matrix C
`R_3` elements of A go to `R_2` fo matrix C
Now, below operations are done on C as follow,
`C_1` elements of C go to `C_3` of B
`C_2` elements of C go to `C_1` of B
`C_3` elements of C go to `C_2` of B
It is found that A = B, then

Options

A. A is symmetric matrix
B. A is an upper triangular matrix
C. A is singular matrix
D. none of these

Answer

To solve the problem step by step, we will first define the matrix \( A \) and then follow the transformations to find matrix \( B \) and verify the conditions given in the question.

### Step 1: Define Matrix \( A \)
Given that the first row of matrix \( A \) is \( [2, -1, -1] \), we can represent matrix \( A \) as follows:
\[ A = \begin{bmatrix} 2 & -1 & -1 \\ a & b & c \\ d & e & f \end{bmatrix} \]
where \( a, b, c, d, e, f \) are the unknown elements of the matrix.

### Step 2: Create Matrix \( C \)
According to the transformations:
- The elements of \( R_1 \) of \( A \) go to \( R_3 \) of \( C \).
- The elements of \( R_2 \) of \( A \) go to \( R_1 \) of \( C \).
- The elements of \( R_3 \) of \( A \) go to \( R_2 \) of \( C \).

Thus, matrix \( C \) can be represented as:
\[ C = \begin{bmatrix} a & b & c \\ d & e & f \\ 2 & -1 & -1 \end{bmatrix} \]

### Step 3: Create Matrix \( B \)
Next, we perform the column transformations on matrix \( C \):
- The elements of \( C_1 \) go to \( C_3 \) of \( B \).
- The elements of \( C_2 \) go to \( C_1 \) of \( B \).
- The elements of \( C_3 \) go to \( C_2 \) of \( B \).

Thus, matrix \( B \) can be represented as:
\[ B = \begin{bmatrix} b & c & a \\ e & f & d \\ -1 & -1 & 2 \end{bmatrix} \]

### Step 4: Set \( A = B \)
Since it is given that \( A = B \), we can equate the corresponding elements of matrices \( A \) and \( B \):
1. From the first row:
- \( 2 = b \)
- \( -1 = c \)
- \( -1 = a \)

2. From the second row:
- \( a = e \)
- \( b = f \)
- \( c = d \)

3. From the third row:
- \( d = -1 \)
- \( e = -1 \)
- \( f = 2 \)

### Step 5: Solve for Unknowns
From the equations we have:
- From \( 2 = b \), we get \( b = 2 \).
- From \( -1 = c \), we get \( c = -1 \).
- From \( -1 = a \), we get \( a = -1 \).
- From \( e = a \), we have \( e = -1 \).
- From \( f = b \), we have \( f = 2 \).
- From \( c = d \), we have \( d = -1 \).

Thus, the complete matrix \( A \) is:
\[ A = \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix} \]

### Step 6: Calculate the Determinant of Matrix \( A \)
To find the determinant of matrix \( A \):
\[ \text{det}(A) = 2 \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} - (-1) \begin{vmatrix} -1 & -1 \\ -1 & 2 \end{vmatrix} + (-1) \begin{vmatrix} -1 & 2 \\ -1 & -1 \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \( \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} = (2)(2) - (-1)(-1) = 4 - 1 = 3 \)
2. \( \begin{vmatrix} -1 & -1 \\ -1 & 2 \end{vmatrix} = (-1)(2) - (-1)(-1) = -2 - 1 = -3 \)
3. \( \begin{vmatrix} -1 & 2 \\ -1 & -1 \end{vmatrix} = (-1)(-1) - (2)(-1) = 1 + 2 = 3 \)

Putting it all together:
\[ \text{det}(A) = 2(3) + 3 - 3 = 6 + 3 - 3 = 6 \]

### Conclusion
The determinant of matrix \( A \) is \( 6 \).