To solve the given problem, we need to evaluate the determinant and find the value of \( k \) in the equation:
\[
\left| \begin{array}{ccc}
a^2 & b^2 & c^2 \\
(a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \\
(a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2
\end{array} \right| = k\lambda \left| \begin{array}{ccc}
a^2 & b^2 & c^2 \\
a & b & c \\
1 & 1 & 1
\end{array} \right|
\]
### Step 1: Expand the Determinant
We start by expanding the determinant on the left-hand side.
\[
\left| \begin{array}{ccc}
a^2 & b^2 & c^2 \\
(a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \\
(a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2
\end{array} \right|
\]
### Step 2: Simplify the Second and Third Rows
We can simplify the second and third rows by performing row operations. Let's denote:
- \( R_2 = (a+\lambda)^2, (b+\lambda)^2, (c+\lambda)^2 \)
- \( R_3 = (a-\lambda)^2, (b-\lambda)^2, (c-\lambda)^2 \)
Now, we can perform the operation \( R_2 - R_3 \):
\[
R_2 - R_3 = \left( (a+\lambda)^2 - (a-\lambda)^2, (b+\lambda)^2 - (b-\lambda)^2, (c+\lambda)^2 - (c-\lambda)^2 \right)
\]
Calculating each term:
\[
(a+\lambda)^2 - (a-\lambda)^2 = 4a\lambda
\]
\[
(b+\lambda)^2 - (b-\lambda)^2 = 4b\lambda
\]
\[
(c+\lambda)^2 - (c-\lambda)^2 = 4c\lambda
\]
Thus, the new second row becomes:
\[
(4a\lambda, 4b\lambda, 4c\lambda)
\]
### Step 3: Update the Determinant
Now, we have:
\[
\left| \begin{array}{ccc}
a^2 & b^2 & c^2 \\
4a\lambda & 4b\lambda & 4c\lambda \\
(a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2
\end{array} \right|
\]
### Step 4: Factor Out Common Terms
We can factor out \( 4\lambda \) from the second row:
\[
= 4\lambda \left| \begin{array}{ccc}
a^2 & b^2 & c^2 \\
a & b & c \\
(a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2
\end{array} \right|
\]
### Step 5: Simplify the Third Row
Next, we can simplify the third row as follows:
\[
R_3 = (a-\lambda)^2, (b-\lambda)^2, (c-\lambda)^2
\]
We can express this as:
\[
R_3 = (a^2 - 2a\lambda + \lambda^2, b^2 - 2b\lambda + \lambda^2, c^2 - 2c\lambda + \lambda^2)
\]
### Step 6: Perform Row Operations Again
Now, we perform another row operation \( R_3 + 2R_2 - R_1 \):
This will help us eliminate the \( a^2, b^2, c^2 \) terms in \( R_3 \).
### Step 7: Calculate the Determinant
After performing these operations, we will find that the determinant simplifies to:
\[
= 4\lambda^3 \left| \begin{array}{ccc}
1 & 1 & 1 \\
a & b & c \\
a^2 & b^2 & c^2
\end{array} \right|
\]
### Step 8: Relate to the Right Side
The determinant on the right-hand side is:
\[
k\lambda \left| \begin{array}{ccc}
a^2 & b^2 & c^2 \\
a & b & c \\
1 & 1 & 1
\end{array} \right|
\]
### Step 9: Equate and Solve for \( k \)
Setting both sides equal gives:
\[
4\lambda^3 = k\lambda
\]
Dividing both sides by \( \lambda \) (since \( \lambda \neq 0 \)):
\[
4\lambda^2 = k
\]
Thus, we find:
\[
k = 4\lambda^2
\]
### Final Answer
The value of \( k \) is:
\[
\boxed{4\lambda^2}
\]
