Let A and B be two matrices of same order 3 x 3 where A=[(1,3, λ+2),(2,4,6),(3,5,8)] B=[(3,2,4),(3,2,5),(2,1,4)]. If matrix Tr(AB)^T+Tr(BA)^T=Tr(BA) then find the value of 2λ?

Given two \(3 \times 3\) matrices \(A\) and \(B\):

\[
A = \begin{pmatrix} 
1 & 3 & \lambda + 2 \\
2 & 4 & 6 \\
3 & 5 & 8 
\end{pmatrix}
\quad \text{and} \quad 
B = \begin{pmatrix} 
3 & 2 & 4 \\
3 & 2 & 5 \\
2 & 1 & 4 
\end{pmatrix}
\]

We are asked to find the value of \(2\lambda\) given the condition:
\[
\text{Tr}((AB)^T) + \text{Tr}((BA)^T) = \text{Tr}(BA)
\]
### Step 1: Understanding the Trace Property
The trace of a matrix is the sum of its diagonal elements. For any matrix \(M\), \(\text{Tr}(M^T) = \text{Tr}(M)\). This is because transposing a matrix does not change its diagonal elements. Therefore, the condition simplifies to:

\[
\text{Tr}(AB) + \text{Tr}(BA) = \text{Tr}(BA)
\]

Subtracting \(\text{Tr}(BA)\) from both sides:

\[
\text{Tr}(AB) = 0
\]

### Step 2: Calculate the Matrix Product \(AB\)
Let's calculate the matrix product \(AB\).

\[
AB = \begin{pmatrix} 
1 & 3 & \lambda + 2 \\
2 & 4 & 6 \\
3 & 5 & 8 
\end{pmatrix}
\cdot
\begin{pmatrix} 
3 & 2 & 4 \\
3 & 2 & 5 \\
2 & 1 & 4 
\end{pmatrix}
\]

Performing the matrix multiplication:

- First row, first column:
\[
1(3) + 3(3) + (\lambda + 2)(2) = 3 + 9 + 2\lambda + 4 = 16 + 2\lambda
\]
- First row, second column:
\[
1(2) + 3(2) + (\lambda + 2)(1) = 2 + 6 + \lambda + 2 = 10 + \lambda
\]
- First row, third column:
\[
1(4) + 3(5) + (\lambda + 2)(4) = 4 + 15 + 4\lambda + 8 = 27 + 4\lambda
\]
- Second row, first column:
\[
2(3) + 4(3) + 6(2) = 6 + 12 + 12 = 30
\]
- Second row, second column:
\[
2(2) + 4(2) + 6(1) = 4 + 8 + 6 = 18
\]
- Second row, third column:
\[
2(4) + 4(5) + 6(4) = 8 + 20 + 24 = 52
\]
- Third row, first column:
\[
3(3) + 5(3) + 8(2) = 9 + 15 + 16 = 40
\]
- Third row, second column:
\[
3(2) + 5(2) + 8(1) = 6 + 10 + 8 = 24
\]
- Third row, third column:
\[
3(4) + 5(5) + 8(4) = 12 + 25 + 32 = 69
\]

Thus, the product \(AB\) is:

\[
AB = \begin{pmatrix} 
16 + 2\lambda & 10 + \lambda & 27 + 4\lambda \\
30 & 18 & 52 \\
40 & 24 & 69 
\end{pmatrix}
\]

### Step 3: Calculate the Trace of \(AB\)
The trace of matrix \(AB\) is the sum of its diagonal elements:

\[
\text{Tr}(AB) = (16 + 2\lambda) + 18 + 69 = 103 + 2\lambda
\]

### Step 4: Solve for \(\lambda\)
From the condition \(\text{Tr}(AB) = 0\):

\[
103 + 2\lambda = 0
\]

Solving for \(\lambda\):

\[
2\lambda = -103
\]

Thus, the value of \(2\lambda\) is \(-103\).

Would you like more details on any of these steps? Here are some related questions you can explore:

1. What are the properties of the trace function in matrix algebra?
2. How does matrix transposition affect the trace of a product?
3. Can you generalize the condition for trace to matrices of higher dimensions?
4. What happens to the trace when you multiply a matrix by a scalar?
5. How does the trace of a product relate to the eigenvalues of the matrices involved?

**Tip:** When multiplying matrices, it’s helpful to label rows and columns to keep track of operations easily.

Explore how to solve a matrix trace condition involving matrices A and B of size 3x3. Understand the steps to find the value of 2λ using matrix algebra concepts and trace properties. Suitable for advanced high school students studying matrix algebra.

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