Question

Let \(\text{A} = \{1, 2, 3, 4, 5\}\) and \(\text{R}\) be a relation defined by \(\text{R} = \{(x, y) : x, y \in \text{A}, x + y = 5\}\). Then, \(\text{R}\) is

• A. reflexive and symmetric but not transitive
• B. an equivalence relation
• C. neither reflexive nor transitive
• D. neither reflexive nor symmetric but transitive

Hint:

When checking properties on small discrete sets, look for the counter-example first. For transitivity, always check paired cycles like \((a,b)\) and \((b,a)\). If their combined reflexive identity \((a,a)\) is missing from the list, you can instantly conclude that the relation is not transitive.

Answer 1

The Correct Option is C

Topic of the Question:
The topic of this question is linear algebra, focusing on the properties of symmetric and skew-symmetric matrices and the rules governing matrix transposes.
Step 1 : Understanding the Question:
We are given two symmetric matrices $A$ and $B$ of the same order. Two new matrices are defined as $X = AB + BA$ and $Y = AB - BA$. We need to find the simplified expression for the transpose of their product, $(\text{XY})^\text{T}$.
Step 2 : Key Formulas and Approach:

Definition of a symmetric matrix: A matrix $M$ is symmetric if $M^\text{T} = M$.

Definition of a skew-symmetric matrix: A matrix $M$ is skew-symmetric if $M^\text{T} = -M$.

Reversal law for the transpose of a product of matrices: $(M_1 M_2)^\text{T} = M_2^\text{T} M_1^\text{T}$.

Distributive property of transposes over addition and subtraction: $(M_1 \pm M_2)^\text{T} = M_1^\text{T} \pm M_2^\text{T}$.

Step 3 : Detailed Explanation:

Since $A$ and $B$ are given as symmetric matrices, we have the relations: $A^\text{T} = A$ and $B^\text{T} = B$.

First, we analyze matrix $X = AB + BA$ by taking its transpose: $X^\text{T} = (AB + BA)^\text{T}$.

Applying the transpose properties: $X^\text{T} = (AB)^\text{T} + (BA)^\text{T} = B^\text{T}A^\text{T} + A^\text{T}B^\text{T}$.

Substituting $A^\text{T} = A$ and $B^\text{T} = B$, we get: $X^\text{T} = BA + AB = AB + BA = X$. Thus, $X$ is a symmetric matrix.

Next, we analyze matrix $Y = AB - BA$ by taking its transpose: $Y^\text{T} = (AB - BA)^\text{T}$.

Applying the transpose properties: $Y^\text{T} = (AB)^\text{T} - (BA)^\text{T} = B^\text{T}A^\text{T} - A^\text{T}B^\text{T}$.

Substituting $A^\text{T} = A$ and $B^\text{T} = B$, we get: $Y^\text{T} = BA - AB = -(AB - BA) = -Y$. Thus, $Y$ is a skew-symmetric matrix.

Now, we find the transpose of the product $XY$ using the reversal law: $(XY)^\text{T} = Y^\text{T} X^\text{T}$.

Substituting our results for $X^\text{T}$ and $Y^\text{T}$ into this expression: $(XY)^\text{T} = (-Y)(X) = -YX$.

Step 4 : Final Answer:
The expression $(\text{XY})^\text{T}$ simplifies to $-YX$, which corresponds to Option (C).