Math Problem Statement
For a given matrix A = [[-1, 3], [1, 2]], find all matrices X such that AX = XA.
Solution
The task is to find all matrices such that for the given matrix . This is a commutative property for the matrix equation.
Step 1: Let .
Step 2: Set up the matrix multiplication for both and .
Compute :
= \begin{pmatrix} -a + 3c & -b + 3d \\ a + 2c & b + 2d \end{pmatrix}$$ Compute $$XA$$: $$XA = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} -1 & 3 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} -a + b & 3a + 2b \\ -c + d & 3c + 2d \end{pmatrix}$$ ### Step 3: Set $$AX = XA$$: $$\begin{pmatrix} -a + 3c & -b + 3d \\ a + 2c & b + 2d \end{pmatrix} = \begin{pmatrix} -a + b & 3a + 2b \\ -c + d & 3c + 2d \end{pmatrix}$$ ### Step 4: Solve the system of equations: Equating the corresponding elements gives: 1. $$-a + 3c = -a + b$$ ⟹ $$3c = b$$ 2. $$-b + 3d = 3a + 2b$$ ⟹ $$-4b + 3d = 3a$$ 3. $$a + 2c = -c + d$$ ⟹ $$a + 3c = d$$ 4. $$b + 2d = 3c + 2d$$ ⟹ $$b = 3c$$ Now solving this system will give all valid matrices $$X$$. Let’s solve these equations systematically. 1. From equation $$3c = b$$, we already know $$b = 3c$$. 2. Substitute $$b = 3c$$ into the second equation: $$-4(3c) + 3d = 3a \quad \Rightarrow -12c + 3d = 3a \quad \Rightarrow d = 4c + a$$ 3. Substitute $$d = 4c + a$$ into the third equation: $$a + 3c = 4c + a \quad \Rightarrow 3c = 4c \quad \text{(True for all $$a, c$$)}.$$ Thus, $$b = 3c$$, and $$d = 4c + a$$. Let $$c = \alpha$$ and $$a = \beta$$ as free variables: $$X = \begin{pmatrix} \beta & 3\alpha \\ \alpha & 4\alpha + \beta \end{pmatrix}$$ This is the general form of all matrices $$X$$ such that $$AX = XA$$. Would you like further clarifications or more examples? --- Here are 5 related questions to consider: 1. How do eigenvalues relate to this matrix commutation property? 2. What happens if $$A$$ is diagonalizable? 3. How do special matrices (e.g., symmetric matrices) affect commutative properties? 4. Can you generalize this result for other 2x2 matrices? 5. How does the solution change for 3x3 matrices? **Tip:** Always check for matrix symmetry and eigenvectors in commutative matrix problems!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Multiplication
Commutative Property of Matrices
Formulas
AX = XA
Matrix multiplication formula
Theorems
Properties of Commutative Matrices
Systems of Linear Equations
Suitable Grade Level
Undergraduate Level (Linear Algebra)