M.Sc. (Semester (CBCS) Examination Nov/Dec-2018
Statistics
LINEAR ALGEBRA
Time: 2½ Hours Max. Marks: 70
Instructions: All Questions carry equal marks.
Figures to the right indicate full marks.
Q.1 Select the correct alternative: 14
If A is an orthogonal matrix, then
A is idempotent AA′ is idempotent
A is skew-symmetric A 0
P 1 1,0 Q 3 11, 12, 13
R 3 3,0 6 S 0
Which of the following sets of vectors are linearly dependent?
Only P P and Q
and R All M
Row space and column space of a matrix
are the subspaces of a single common vector space
always coincide
are not vector spaces
have the same dimension
A single null vector
forms a linearly dependent set of vectors
forms a linearly independent set of vectors
does not form a vector space
forms a vector space called null space
If matrix G is a g-inverse of matrix then
H GAG is a g-inverse of A such that rank rank
H GAG is a g-inverse of A such that rank rank
H GAG is a g-inverse of A such that rank rank
H GAG is not a g-inverse of A
Let A be a 3 x 3 matrix. A necessary and sufficient condition for existence of
a non-trivial solution to the system of linear equations 0 is
rank A 3 rank A 3
rank A 2 rank A 2
The maximum number of linearly independent solutions to a system of linear
equations where is a 3 × 5 matrix of rank 3 is
0 1
2 3
Elementary column operation
changes rank of a matrix
is essentially post-multiplying the given matrix by an elementary matrix
is essentially pre-multiplying the given matrix by an identity matrix
is essentially post-multiplying the given matrix by an orthogonal matrix
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SLR-VR-473
The eigen values of a triangular matrix are
the diagonal elements of the matrix
the off-diagonal elements of the matrix
zero and one
none of C
10) The eigen values of 2 × 2 matrix A are 2 and 6. Then
8 12
12 4
11) If A is a singular matrix of order 3. Then its eigen value is
0 1
2 3
12) The characteristic polynomial of matrix A is λ4 − 1. Then, A4
cannot be known I
A λA
13) The quadratic form is
positive definite negative definite
positive semi-definite negative semi-definite
14) Let A be a real symmetric matrix of order 4 and rank then the
number of non-zero eigen value of A is
1 2
3 4
Q.2 Answer the following. (Any four) 08
Define vector subspace. Give an example.
Determine any two g-inverses of a 5].
If is an eigen value of a nonsingular matrix show that is an
eigen value of A-1.
Let p and q be the numbers of positive and negative in the quadratic
form Q 2
1
i
n
i
i x d

then what is a necessary and sufficient condition for Q
to be positive definite?
State the matrix associated with the quadratic form −
Write notes on. (Any two) 06
Elementary matrix operations
Moore-Penrose inverse
Characteristic value problem
Q.3 Answer the following. (Any two) 08
Prove that any superset of a linearly dependent set of vectors is linearly
dependent.
If G is g-inverse of matrix show that G1 GAG is also a g-inverse of
A.
Show that the eigen values of an idempotent matrix are either 0 or 1.
Answer the following. (Any one) 06
Describe Gram-Scmidt process of orthogonalisation.
Prove that the definiteness of a quadratic form is invariant under
nonsingular linear transformation.
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Q.4 Answer the following. (Any two) 10
If A and B are square matrices of order show that rank rank
rank
Let A and B be the two matrices such that the products AB and BA are
well defined. Show that AB and BA have the same set of eigen values.
Reduce the following quadratic form to a form containing only square
terms.
2
2
Answer the following. (Any one) 04
Express matrix
2 4
1 3
as the sum of a symmetric and a skew-symmetric
matrices.
Show that all nonsingular matrices of order have the same rank.
Q.5 Answer the following. (Any two) 14
Show that the rank of sum of two matrices cannot exceed sum of their ranks.
State and prove Caley-Hamilton theorem.
State and prove a necessary and sufficient condition for a real quadratic
form to be positive definite.