The matrix `A={:[(lamda_(1)^(2),lamda_(1)lamda_(2),lamda_(1)lamda_(3)),(lamda_(2)lamda_(1),lamda_(2)^(2),lamda_(2)lamda_(3)),(lamda_(3)lamda_(1),lamda_(3)lamda_(2),lamda_(3)^(2))]:}` is idempotent if `lamda_(1)^(2)+lamda_(2)^(2)+lamda_(3)^(2)=k` where `lamda_(1),lamda_(2),lamda_(3)` are non-zero real numbers. Then the value of `(10+k)^(2)` is . . .

Answer

To determine the value of \((10 + k)^2\) for the given idempotent matrix \( A \), we will follow these steps:

### Step 1: Understand the Definition of Idempotent Matrix
An idempotent matrix \( A \) satisfies the condition:
\[ A^2 = A \]
This means that when the matrix is multiplied by itself, the result is the matrix itself.

### Step 2: Calculate \( A^2 \)
Given the matrix:
\[ A = \begin{pmatrix} \lambda_1^2 & \lambda_1 \lambda_2 & \lambda_1 \lambda_3 \\ \lambda_2 \lambda_1 & \lambda_2^2 & \lambda_2 \lambda_3 \\ \lambda_3 \lambda_1 & \lambda_3 \lambda_2 & \lambda_3^2 \end{pmatrix} \]
We need to compute \( A^2 \). The element at position (1,1) in \( A^2 \) can be calculated as:
\[ (\lambda_1^2)(\lambda_1^2) + (\lambda_1 \lambda_2)(\lambda_2 \lambda_1) + (\lambda_1 \lambda_3)(\lambda_3 \lambda_1) = \lambda_1^4 + \lambda_1^2 \lambda_2^2 + \lambda_1^2 \lambda_3^2 \]
Continuing this for all elements of \( A^2 \), we can express \( A^2 \) in terms of \( \lambda_1, \lambda_2, \lambda_3 \).

### Step 3: Set \( A^2 = A \)
After calculating \( A^2 \), we will set it equal to \( A \) and compare corresponding elements. This will yield a system of equations.

### Step 4: Derive the Equation
From the idempotent condition, we will derive the equation:
\[ \lambda_1^2 + \lambda_2^2 + \lambda_3^2 = k \]
This implies that the sum of the squares of the eigenvalues must equal \( k \).

### Step 5: Solve for \( k \)
From the derived equations, we will find that:
\[ k = 1 \]
This is because we need the sum of the squares of the eigenvalues to equal a constant.

### Step 6: Calculate \( (10 + k)^2 \)
Now that we have found \( k = 1 \), we can substitute this value into the expression:
\[ (10 + k)^2 = (10 + 1)^2 = 11^2 = 121 \]

### Final Answer
Thus, the value of \( (10 + k)^2 \) is:
\[ \boxed{121} \]