1 1 0
1 0 1
0 0 1
1 1 0
.
For matrix A:
(a) Find row echelon form of A through Gaussian Elimination for solving Ax = b, and based
on the value of a
b =a110,
determine different states of the system. For what values of a the system is definite, for
what values of a the system is inconsistent, and for what values of a the system is indefinite.
(b) Find rank(A).
(c) What are the pivots
(d) Is the matrix invertible
(e) What is dim(N(A))
[8+3+3+3+3= 20 marks]
SIT787 Math for AI Assignment 2 2020 Tri 1 Page 2 of 3
Now consider
A =3 11 31 1
for questions 2, 3, 4, and 5.
Question 2) Find S = AT A matrix and
(a) Find the characteristic polynomial of S
(b) Find the eigenvalues of S
(c) Find the eigenvectors S.
(d) Are the eigenvectors orthonormal If they are not, convert them into orthobnormal using
Gram-Schmidt process. Call them →v1,→v2.
(e) Consider λ1 ≥ λ2.Set σ1 =√λ1 and σ2 =√
λ2 and make this matrix
D =σ1 00 σ20 0 .
Make a matrix V with the orthonormal columns you obtained cor
responding to the order
of eigenvalues V =→v1→v2. Show that V is an orthogonal matrix.
[2+2+2+8+3+3= 20 marks]
Question 3) Find T = AAT matrix and
(a) Find the characteristic polynomial of T
(b) Find the eigenvalues of T
(c) Find the eigenvectors T.
(d) Are the eigenvectors orthonormal If they are not, convert them into orthobnormal using Gram-Schmidt process. Call them →x1,→x2,→x3.
[2+3+3+9+3= 20 marks]
SIT787 Math for AI Assignment 2 2020 Tri 1 Page 3 of 3
Question 4) Consider two orthonormal vectors you obtained in question 2.
(a) Using them we want to make three orthonormal vectors →u1,→u2,→u3 such that
→u1 =1σ1A→v1
→u2 =1σ2A→v2
Also, find the third vector →u3,
→u3 =
abc
such that →u3 is a unit vector and orthogonal to →u1 and →u2. In other words
||→u3||2 = 1 and →u3 ⊥
→u1 and →u3 ⊥
→u2.
Put these three vectors in a matrice U =
→u1
→u2
→u3.
(b) Show that U is an orthogonal matrix.
(c) Compute UDV T.
(d) What you can say about UDV T