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 determine the value of \( \lambda_1^2 + \lambda_2^2 + \lambda_3^2 \) given that the matrix

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

is idempotent, meaning \( A^2 = A \).

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

To find \( A^2 \), we perform the matrix multiplication \( A \times A \):

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

### Step 2: Compute the elements of \( A^2 \)

Calculating the (1,1) element:

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

Calculating the (1,2) element:

\[ \lambda_1^2 \cdot \lambda_1 \lambda_2 + \lambda_1 \lambda_2 \cdot \lambda_2^2 + \lambda_1 \lambda_3 \cdot \lambda_3 \lambda_2 = \lambda_1^2 \lambda_2 + \lambda_1 \lambda_2^3 + \lambda_1 \lambda_2 \lambda_3^2 \]

Calculating the (1,3) element:

\[ \lambda_1^2 \cdot \lambda_1 \lambda_3 + \lambda_1 \lambda_2 \cdot \lambda_2 \lambda_3 + \lambda_1 \lambda_3 \cdot \lambda_3^2 = \lambda_1^2 \lambda_3 + \lambda_1 \lambda_2 \lambda_3^2 + \lambda_1 \lambda_3^3 \]

Continuing this process for all elements, we can express \( A^2 \) in terms of \( \lambda_1, \lambda_2, \lambda_3 \).

### Step 3: Set \( A^2 = A \)

After calculating all elements of \( A^2 \), we equate them to the corresponding elements of \( A \). This will yield a system of equations involving \( \lambda_1, \lambda_2, \lambda_3 \).

### Step 4: Solve the system of equations

From the equations obtained, we can derive relationships between \( \lambda_1, \lambda_2, \lambda_3 \). Specifically, we will find that:

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

### Conclusion

Thus, the value of \( \lambda_1^2 + \lambda_2^2 + \lambda_3^2 \) is:

\[ \boxed{1} \]