Essential Concepts in Linear Algebra and Logic

Sure, I'll address each of the directives.

**Q-1**

(1) **Define Symmetric Matrix:**
A symmetric matrix is a square matrix that is equal to its transpose. Mathematically, a matrix \( A \) is symmetric if \( A = A^T \), where \( A^T \) denotes the transpose of \( A \). This implies that \( a_{ij} = a_{ji} \) for all elements in the matrix.

(2) **Define Tautology:**
A tautology is a statement or formula that is always true, regardless of the truth values of its component propositions. In logical terms, a tautology is a proposition that evaluates to true in every possible interpretation.

**Reh (P.T.O)**

(3) **Write down the condition for consistency of a non-homogeneous system of linear equations:**
A non-homogeneous system of linear equations \( AX = B \) is consistent if and only if the augmented matrix \([A|B]\) has the same rank as the coefficient matrix \( A \). In other words, \(\text{rank}(A) = \text{rank}([A|B])\).

(4) **Negate the statement: London is a city:**
The negation of the statement "London is a city" is "London is not a city."

**Find the cofactor of \( a_{11} \) in the matrix:**
Given the matrix:
\[ \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{23} & a_{23} & a_{23} \\
a_{31} & a_{22} & a_{33}
\end{bmatrix} \]

To find the cofactor of \( a_{11} \):
1. Remove the row and column containing \( a_{11} \):
\[ \begin{bmatrix}
a_{23} & a_{23} \\
a_{22} & a_{33}
\end{bmatrix} \]

2. Calculate the determinant of the 2x2 matrix:
\[ \text{det} \left( \begin{bmatrix}
a_{23} & a_{23} \\
a_{22} & a_{33}
\end{bmatrix} \right) = a_{23}a_{33} - a_{23}a_{22} = a_{23}(a_{33} - a_{22}) \]

3. The cofactor of \( a_{11} \) is:
\[ \text{Cofactor}_{a_{11}} = (-1)^{1+1} \cdot (a_{23}(a_{33} - a_{22})) = a_{23}(a_{33} - a_{22}) \]

**Q-2(a)** (Any Five)

(1) **State whether the following statement is false with reason: "(N, +) is a monoid."**
This statement is **true**, not false. The set of natural numbers \( \mathbb{N} \) with the operation of addition \( + \) forms a monoid because it satisfies the following properties:
- **Closure:** For any \( a, b \in \mathbb{N} \), \( a + b \in \mathbb{N} \).
- **Associativity:** For any \( a, b, c \in \mathbb{N} \), \( (a + b) + c = a + (b + c) \).
- **Identity element:** The identity element for addition in \( \mathbb{N} \) is 0, since \( a + 0 = 0 + a = a \).

(2) **Define Zero Divisor with an example:**
A zero divisor in a ring \( R \) is a non-zero element \( a \in R \) such that there exists a non-zero element \( b \in R \) where \( ab = 0 \). 
**Example:** In the ring \( \mathbb{Z}_6 \) (integers modulo 6), the element 2 is a zero divisor because \( 2 \times 3 \equiv 0 \mod 6 \), and neither 2 nor 3 is zero in \( \mathbb{Z}_6 \).