[Solved] SIT292 Exam

1. Consider Hasse diagram
   • List the pairs of elements that are not related via this partial order.
   • Construct, as far as possible, the table of least upper bounds for pairs of elements under this partial order.
   • Is this poset a lattice? (provide an argument for your answer).
   • Let p b), (a, b), (b, a), (c, c), (c, d), (d, c)} be a relation on the set {a, b,c, d}, Construct p2, list explicitly all the ordered pairs that form p2.
   • Draw a directed graph diagram for p and p2. Which ordered pair(s) must be adjoined to p² to complete it into an txluivalence relation on the set {a, b, c, d}?

2. For the following system of linear
   • Use Gaussian elimination to put the system into row echelon form
   • Use Gauss-Jordan elimination to change your row echelon form into reduced row echelon form.
   • Solve the system of equations using either your answer to part (a) or your answer to part
   • Verify that your solutions satisfy this system of equations.

= IS marks)

3. Consider matrix

3

1

2

• Confirm that the matrix A is invertible.
• Find all the cofactors Cij of the matrix A and hence find adj(A).
• Verify that the adj(A) you obtained is correct by multiplying it with A.

4. For the following matrix
   • Find the eigenvalues (one eigenvalue is Al = 2)
   • For each eigenvalue find the corresponding eigenvector(s).
   • Determine if possible a matrix p so that B = pIAP is in diagonal form. Write down B and p.

(2 + S F 6 = 16 marks)

5. The vector space of solutions of Ax = O

is generated by

• Verify that each vector is a solution.
• Show that any solution can be written as a linear combination

for a suitable choice Of q , 02, c].

• What is the dimension of the row-space of A and the dimension of its nullspace? State one basis for the nullspace of A.

6. For matrix
   • Determine the row-rank.
   • Find a set of generators for the row space of A.
   • State one possible basis for the row space of A and explain why it is a basis.

7. The 6-tuples:

= 011100, u2 = 111010, = 110011 form a biLSis for a (6,3) linear code.

• Write down the generator matrix for this code.
• Construct code words for the message blocks: 101, 110, 011, 010.
• Construct the parity check matrix for this code.
• Decode if possible