##### Matrices
• Idempotent Matrix : For a matrix A , if we get A2 = A , then it is an idempotent matrix.
• Nilpotent Matrix : For a matrix A, if at some m we get Am = 0, then it is called nilpotent matrix, and smallest value of m at which this condition is satisfied is called order of matrix.
• Involutory Matrix : For a matrix A , if we get A2 = I , then it is called involutory matrix.
• Singular Matrix : For a matrix A, if det(A) is zero, then it is called singular matrix.
• Non- Singular Matrix: For a matrix A , if det(A) is non-zero , then it is called non - singular matrix.
• Symmetric matrix : For a matrix , if AT = A , then it is symmetric matrix.
• Skew- Symmetric matrix : For a matrix , if AT = -A , then it is skew - symmetric matrix.
• Sum of Symmetric and skew symmetric matrix is twice the matrix.
##### Petersan Graph
• Petersan graph is a graph with 10 vertices and 15 edges.
• It is a non-planar graph.
• It has got hamiltonian path but not hamiltonain cycle.
• It has got chromatic number 3.
set2018
29 Aug 2017 06:01 pm

pls explain point 3 .what are the condition for checking hamilton path

Akshay Saxena
30 Aug 2017 10:22 am

start from a vertex and you can trace each vertex exactly once without repeating vertices but you cant reach to the starting vertex without repeating try it.

29 Aug 2017 07:40 pm

One more point that can be added is this graph contains K5 and its isomorphic image as well..

##### Mathematical Logic : Rules of Inference
• Precedence of logical operators is as follows:
¬  > ∧  >  ∨   > →  >  ↔
where, ¬  , ∧  ,  ∨  is left associative and → ,  ↔ is right associative.
Example: P →Q →R can be wriiten as  (P →(Q →R))
• Following are the rules of inference:
1. Modus Ponens:
P
P
________
Q
________
2. Modus Tollens:
¬Q
P->Q
________
¬P
________
3. Disjunctive Syllolgism:
P∨Q
¬P
________
Q
________
4. Constructive Dilemma:
(PQ ) ∧ (RS )
(P∨R )
________
Q∨S
________
5. Destructive Dilemma:
(PQ ) ∧ (RS )
(¬Q∨¬S)
________
¬P∨¬R
________
6. Disjunctive Syllolgism:
P∨Q
¬P
________
Q
________

Example:
If child studies , mom will not scould
Mom scoulded
Inference: Child did not study.
Ans. p =  child studies
q = Mom scould
p->~q
q
_____
~p          ///using Modus Tollens
_____
~p means child did not study.
Thus , given inference is true.