M.Sc.(Statistics) (Semester (CBCS) Examination, 2017
LINEAR ALGEBRA
Day Date: Saturday, 22-04-2017 Max. Marks: 70
Time: 10:30 AM to 01.00 PM
N.B. Q. No. and Q. No are compulsory.
Attempt any three from Q. No. to Q. No.
Figures to the right indicate full marks.
Q.1 Choose correct alternative:
The maximum number of linearly independent vectors in
vector space is
Rank of vector space Dimension of vector space
Power set None of these
2 The g-inverse of matrix
is





None of these
3 The quadratic form
is
Positive definite Negative definite
Indefinite Positive semi- definite
4 The vector is unit vector when
All elements are unity All elements are zero
element is unit None of these
Rank of matrix
is
4 2 3 1
Fill in the blanks.
The dimension of is
The scalar product of two orthogonal vectors is
exists if and only if A is
Rank of every non-singular matrix of order n is
If
then
Page 2 of 2
State the following sentence are True or False:
P is orthogonal matrix if .
Every subspace of a vector space is also a vector space.
If is an idempotent matrix then trace Rank .
A real quadratic form is invariant under non-singular linear
transformation.
Q.2 Define the terms: 6+8
Vector space
Linear dependence
Row space and column space
Write short notes on the following:
System of linear equations
Spectral decomposition
Q.3
Examine the definiteness of 7+7
Prove that for a square matrix row rank is same as column rank.
Q.4 Verify the Caley-Hamilton theorem for
also find and
.
8+6
Show that eigen values of an idempotent are either zero or zone.
Q.5 Explain the following: 8+6
Singular value decomposition
Cholesky decomposition
Show that, the system of equations is consistent if and
only if rank rank
Q.6

Let be a 2X2 matrix and let be the
characteristic polynomial of then show that
7+7
Let A be a n n matrix. Prove that is singular if and only if zero
is its eigen value.
Q.7 State and prove necessary and sufficient condition for positive
definiteness of quardratic form.
7+7
Explain Gramm-schmidt process of orthogonalization.