M.Sc. (Statistics) (Semester (CBCS) Examination, 2017
LINEAR ALGEBRA
Day Date: Thursday, 20-04-2017 Max. Marks: 70
Time: 10.30 AM to 01.00 PM
N.B. Q.1 and Q.2 are compulsory.
Attempt five questions.
Attempt any three questions from Q. 3 to 7.
Figures to the right indicate full marks.
Q.1 Select the correct alternative: 10
The dimension of a vector space generated by the columns
of a nonsingular matrix A of order 3
3 3-rank(A) rank(A)
is an eigen value of every singular matrix.
1 0 The order of the matrix
If G is a g-inverse of then
rank rank rank rank
rank rank rank rank
A positive definite matrix has
some positive and some negative eigen values
0 as its eigen value
All nonnegative eigen values
All positive eigen values
The quadratic form is
Positive definite Negative definite
Positive semi-definite Negative semi-definite
Q.1 Fill in the blanks: 05
A vector space is closed under the operations of and

The eigen value of a square matrix A of order 3 are 2 and
3 then rank
The inverse of is
The elementary matrix which if post multiplied by a 3x2
matrix interchanges the first and third rows of A is
A system of linear equations Ax b in n variables has unique
solution if and only if
Page 2 of 3
Q.1 State true and false 04
A single null vector always forms a linearly dependent set of
vectors.
A real symmetric matrix of order n has n orthogonal eigen
vectors.
A homogeneous system of equations is not necessarily
consistent.
There exist some nonsingular linear transformations that
change the definiteness of any quadratic form
Q.2 Show that the representation of any vector in terms of the
basic vectors in unique.
03
ii) Prove that a nonsingular matrix has a unique inverse. 03
Write short notes on the following. 08
Singular values of decomposition
ii) Classification of quadratic forms
Q.3 Show that every basis for n-dimensional Euclidean space
contains exactly n vectors.
07
Prove that row rank of a matrix is same as its column rank. 07
Q.4 State and prove Caley-Hamilton theorem. 07
Describes a procedure of obtaining a G inverse of a matrix and
using that obtain a G inverse of

07
Q.5
Obtain spectral decomposition of A
√
√
and hence find
A6.
07
Show that 07
The characteristic roots of a triangular matrix are the
diagonal elements of that matrix
ii) If A and B are the two matrices such that the products AB
and BA are well defined, AB and BA have the same set of
eigen values.
Q.6 State and prove a necessary and sufficient condition for a
quadratic form to be positive definite.
07
Show that the maximum number of linearly independent solutions
to the system of equations Ax 0 is n-rank where n is the
number of columns in A.
07
Q.7 Reduce the following quadratic form to a form containing only
square terms.
Let be the characteristic roots of an matrix A.
Show that
and trace Σ
Page 3 of 3
Q.4 Prove that subgroup of Nilpotent group is Nilpotent. 07
State and prove Second Sylow Theorem. 07
Q.5 State and prove division algorithm. 07
Let and be in Z5 where f 4x3 4x2 3x 3 and
4x2 Then show that divides
07
Q.6 State and prove Eisenstein Criteria for irreducibility over Q. 07
Show that a group of order 255 is not simple. 07
Q.7 Let X be any G-set, then prove that for any 07
Let be the characteristic roots of an matrix A.
Show that
and trace Σ
s