If the matrix `A = [[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)]]` is idempotent,
the value of `lambda_(1)^(2) + lambda_(2)^(2) + lambda _(3)^(2)` is

Explanation

To solve the problem, we need to find the value of \( \lambda_1^2 + \lambda_2^2 + \lambda_3^2 \) given that the matrix \( A \) is idempotent. An idempotent matrix satisfies the condition \( A^2 = A \).

Given the matrix:

\[ A = \begin{bmatrix} \lambda_1^2 & \lambda_1 \lambda_2 & \lambda_1 \lambda_3 \\ \lambda_2 \lambda_1 & \lambda_2^2 & \lambda_2 \lambda_3 \\ \lambda_3 \lambda_1 & \lambda_3 \lambda_2 & \lambda_3^2 \end{bmatrix} \]

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we will multiply the matrix \( A \) by itself.

\[ A^2 = A \cdot A \]

Calculating the elements of \( A^2 \):

- The element at position (1,1):
\[ \lambda_1^2 \cdot \lambda_1^2 + \lambda_1 \lambda_2 \cdot \lambda_2 \lambda_1 + \lambda_1 \lambda_3 \cdot \lambda_3 \lambda_1 = \lambda_1^4 + \lambda_1^2 \lambda_2^2 + \lambda_1^2 \lambda_3^2 \]

- The element at position (1,2):
\[ \lambda_1^2 \cdot \lambda_1 \lambda_2 + \lambda_1 \lambda_2 \cdot \lambda_2^2 + \lambda_1 \lambda_3 \cdot \lambda_3 \lambda_2 = \lambda_1^2 \lambda_1 \lambda_2 + \lambda_1 \lambda_2^3 + \lambda_1 \lambda_2 \lambda_3^2 \]

- The element at position (1,3):
\[ \lambda_1^2 \cdot \lambda_1 \lambda_3 + \lambda_1 \lambda_2 \cdot \lambda_2 \lambda_3 + \lambda_1 \lambda_3 \cdot \lambda_3^2 = \lambda_1^2 \lambda_1 \lambda_3 + \lambda_1 \lambda_2 \lambda_2 \lambda_3 + \lambda_1 \lambda_3^3 \]

Continuing this process for all elements of \( A^2 \), we will find:

\[ A^2 = \begin{bmatrix} \lambda_1^4 + \lambda_1^2 \lambda_2^2 + \lambda_1^2 \lambda_3^2 & \lambda_1^2 \lambda_2 + \lambda_2^3 + \lambda_2 \lambda_3^2 & \lambda_1^2 \lambda_3 + \lambda_2 \lambda_3 + \lambda_3^3 \\ \text{(similar calculations for other rows)} \end{bmatrix} \]

### Step 2: Set \( A^2 = A \)

Since \( A \) is idempotent, we set \( A^2 = A \). This gives us a system of equations based on the equality of corresponding elements in the matrices.

For example, from the (1,1) position:

\[ \lambda_1^4 + \lambda_1^2 \lambda_2^2 + \lambda_1^2 \lambda_3^2 = \lambda_1^2 \]

Rearranging gives:

\[ \lambda_1^4 + \lambda_1^2 (\lambda_2^2 + \lambda_3^2 - 1) = 0 \]

### Step 3: Solve the equations

We can derive similar equations for all positions. The key observation is that if \( A \) is idempotent, the eigenvalues of \( A \) must be either 0 or 1.

Thus, we can conclude that:

\[ \lambda_1^2 + \lambda_2^2 + \lambda_3^2 = 1 \]

### Final Answer

The value of \( \lambda_1^2 + \lambda_2^2 + \lambda_3^2 \) is:

\[ \boxed{1} \]