If matrix A `=[{:( a,b,c),(b,c,a),(c,a,b) :}]` where a,b,c are real positive number ,abc =1 and `A^(T) A = I ` then
Which of the following is/are true

Options

A. `a+b+c= 1`
B. `a^(2)+b^(2) +c^(2) =1`
C. ` ab+bc+cane0`
D. `a^(3) +b^(3) +c^(3)= 4`

Answer

To solve the problem, we need to analyze the given matrix \( A \) and the conditions provided.

### Step 1: Understand the matrix \( A \)
The matrix \( A \) is given as:
\[ A = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} \]
where \( a, b, c \) are positive real numbers and \( abc = 1 \).

### Step 2: Calculate \( A^T \)
The transpose of matrix \( A \) is:
\[ A^T = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} \]
Since the matrix is symmetric, \( A^T = A \).

### Step 3: Calculate \( A^T A \)
Now, we compute \( A^T A \):
\[ A^T A = A A = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} \]
Calculating the elements of \( A^T A \):
- The (1,1) entry:
\[ a^2 + b^2 + c^2 \]
- The (1,2) entry:
\[ ab + bc + ca \]
- The (1,3) entry:
\[ ac + ab + bc \]
- The (2,1) entry:
\[ ab + bc + ca \]
- The (2,2) entry:
\[ b^2 + c^2 + a^2 \]
- The (2,3) entry:
\[ bc + ca + ab \]
- The (3,1) entry:
\[ ac + ab + bc \]
- The (3,2) entry:
\[ bc + ca + ab \]
- The (3,3) entry:
\[ c^2 + a^2 + b^2 \]

Combining these, we have:
\[ A^T A = \begin{pmatrix} a^2 + b^2 + c^2 & ab + bc + ca & ab + ac + bc \\ ab + bc + ca & b^2 + c^2 + a^2 & ab + ac + bc \\ ab + ac + bc & ab + ac + bc & c^2 + a^2 + b^2 \end{pmatrix} \]

### Step 4: Set \( A^T A = I \)
Since \( A^T A = I \), we have:
\[ \begin{pmatrix} a^2 + b^2 + c^2 & ab + bc + ca & ab + ac + bc \\ ab + bc + ca & b^2 + c^2 + a^2 & ab + ac + bc \\ ab + ac + bc & ab + ac + bc & c^2 + a^2 + b^2 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 5: Equate the entries
From the equality, we can derive:
1. \( a^2 + b^2 + c^2 = 1 \)
2. \( ab + bc + ca = 0 \)

### Step 6: Analyze \( a + b + c \)
Using the identity:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
Substituting the known values:
\[ (a + b + c)^2 = 1 + 2(0) = 1 \]
Thus, \( a + b + c = 1 \).

### Step 7: Calculate \( a^3 + b^3 + c^3 \)
Using the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
Substituting the known values:
\[ a^3 + b^3 + c^3 - 3(1) = 1(1 - 0) \Rightarrow a^3 + b^3 + c^3 - 3 = 1 \]
Thus, \( a^3 + b^3 + c^3 = 4 \).

### Conclusion
From the above calculations, we can conclude:
1. \( a^2 + b^2 + c^2 = 1 \) (True)
2. \( ab + bc + ca = 0 \) (True)
3. \( a + b + c = 1 \) (True)
4. \( a^3 + b^3 + c^3 = 4 \) (True)