Algebraic Structures | Question-4: Matrix Commutator Problem
Question
Let A be the set of all 2 × 2 real matrices. Consider the specific matrix:
\[
M = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}
\]
Definition of Set B
The set B is defined as:
\[
B = \{ X \in A \mid MX = XM \}
\]
This defines the set of all matrices that commute with matrix M.
Question: If we let \( X = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \), what conditions must p, q, r, s satisfy for X to belong to B? That is, when does \( MX = XM \)?
Solution
Understanding Matrix Commutation
Two matrices commute if their product is the same regardless of order:
\[
MX = XM
\]
To find all matrices X that commute with M, we need to solve this equation for X.
Step 1: Multiply Both Ways
First, compute the product \( MX \):
\[
MX = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} p+r & q+s \\ r & s \end{pmatrix}
\]
Then compute the product \( XM \):
\[
XM = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} p & p+q \\ r & r+s \end{pmatrix}
\]
Step 2: Set Products Equal
For \( X \) to commute with \( M \), we require \( MX = XM \). Setting the matrices equal gives:
\[
\begin{pmatrix} p+r & q+s \\ r & s \end{pmatrix} = \begin{pmatrix} p & p+q \\ r & r+s \end{pmatrix}
\]
Important Note: Two matrices are equal if and only if all their corresponding entries are equal.
Step 3: Compare Corresponding Entries
By comparing corresponding matrix entries, we obtain the following system of equations:
Entry-by-Entry Comparison
\[
\begin{aligned}
&(1,1): \quad p + r = p \\
&(1,2): \quad q + s = p + q \\
&(2,1): \quad r = r \\
&(2,2): \quad s = r + s
\end{aligned}
\]
Simplifying these equations:
- Top-left: \( p + r = p \) gives \( \boxed{r = 0} \)
- Top-right: \( q + s = p + q \) gives \( \boxed{s = p} \)
- Bottom-left: \( r = r \) is automatically satisfied
- Bottom-right: \( s = r + s \) is consistent with \( r = 0 \)
Step 4: General Form of Commuting Matrices
From the conditions \( r = 0 \) and \( s = p \), every matrix that commutes with \( M \) must have the form:
\[
X = \begin{pmatrix} p & q \\ 0 & p \end{pmatrix}
\]
where \( p \) and \( q \) can be any real numbers.
Key Observation: These matrices form a 2-dimensional vector space parameterized by p and q.
Final Solution
The necessary and sufficient conditions for \( X \) to commute with \( M \) are:
\[
\boxed{r = 0 \quad \text{and} \quad s = p}
\]
Therefore, the set \( B \) consists of all matrices of the form:
\[
B = \left\{ \begin{pmatrix} p & q \\ 0 & p \end{pmatrix} \mid p, q \in \mathbb{R} \right\}
\]
\[
X = \begin{pmatrix} p & q \\ 0 & p \end{pmatrix}
\]