Math Problem Statement
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?
Solution
Given two matrices and :
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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Matrix Multiplication
Matrix Trace
Formulas
Trace of a Matrix
Theorems
-
Suitable Grade Level
Advanced High School