Algebraic Structures | Question-1
Problem Statement:
Let \( A \) be the set of \( 2 \times 2 \) matrices with real number entries.
Given:
\[M = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\]
\[B = \{ X \in A \mid MX = XM \}\]
Determine which of the following matrices in \( A \) belong to \( B \):
Solution
Define the general form of \( X \):
Let \( X = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), where \( a, b, c, d \in \mathbb{R} \).
We need to find conditions on \( a, b, c, d \) such that \( MX = XM \).
Compute \( MX \) and \( XM \):
Compute \( MX \):
\[ MX = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a + c & b + d \\ c & d \end{pmatrix} \]
Compute \( XM \):
\[ XM = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a & a + b \\ c & c + d \end{pmatrix} \]
Set \( MX = XM \) and derive conditions:
Equality Conditions:
From \( MX = XM \), we have:
\[ \begin{pmatrix} a + c & b + d \\ c & d \end{pmatrix} = \begin{pmatrix} a & a + b \\ c & c + d \end{pmatrix} \]
Equating corresponding entries gives:
- \( a + c = a \Rightarrow c = 0 \)
- \( b + d = a + b \Rightarrow d = a \)
- \( c = c \) (automatically satisfied)
- \( d = c + d \Rightarrow c = 0 \) (already satisfied)
Therefore, \( X \) commutes with \( M \) if and only if:
\[ \boxed{c = 0 \quad \text{and} \quad d = a} \]
The general form of matrices in \( B \) is:
\[ X = \begin{pmatrix} a & b \\ 0 & a \end{pmatrix}, \quad a,b \in \mathbb{R} \]
Test each given matrix against the conditions:
We check each matrix for \( c = 0 \) and \( d = a \):
Matrix 1: \( \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \)
\( a=1, b=1, c=0, d=1 \)
Check: \( c=0 \) ✅, \( d=a \) ✅ (1 = 1)
✅ Belongs to \( B \)
Matrix 2: \( \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \)
\( a=1, b=1, c=1, d=1 \)
Check: \( c=0 \) ❌ (1 ≠ 0)
❌ Does not belong to \( B \)
Matrix 3: \( \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \)
\( a=0, b=0, c=0, d=0 \)
Check: \( c=0 \) ✅, \( d=a \) ✅ (0 = 0)
✅ Belongs to \( B \)
Matrix 4: \( \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \)
\( a=1, b=1, c=1, d=0 \)
Check: \( c=0 \) ❌ (1 ≠ 0)
❌ Does not belong to \( B \)
Matrix 5: \( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)
\( a=1, b=0, c=0, d=1 \)
Check: \( c=0 \) ✅, \( d=a \) ✅ (1 = 1)
✅ Belongs to \( B \)
Matrix 6: \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
\( a=0, b=1, c=1, d=0 \)
Check: \( c=0 \) ❌ (1 ≠ 0)
❌ Does not belong to \( B \)
Conclusion
A matrix \( X = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) belongs to \( B \) if and only if it satisfies:
\[ c = 0 \quad \text{and} \quad d = a \]
Final Answer:
✅ Matrices in \( B \):
- \( \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \)
- \( \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \)
- \( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)
❌ Matrices not in \( B \):
- \( \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \)
- \( \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \)
- \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
Summary: Out of the 6 given matrices, 3 satisfy the commuting condition \( MX = XM \) and belong to \( B \).