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QUESTION: 1

The system of linear equations

4x + 2y = 7

2x + y = 6 has

Solution:

(b) This can be written as AX = B Where A

Angemented matrix

rank(A) ≠ rank(). The system is inconsistant .So system has no solution.

QUESTION: 2

For the following set of simultaneous equations:

1.5x – 0.5y = 2

4x + 2y + 3z = 9

7x + y + 5z = 10

Solution:

(a)

∴ rank of() = rank of(A) = 3

∴ The system has unique solution.

QUESTION: 3

The following set of equations has

3 x + 2 y + z = 4

x – y + z = 2

-2 x + 2 z = 5

Solution:

(b)

∴ rank (A) = rank () = 3

∴ The system has unique solution

QUESTION: 4

Consider the system of simultaneous equations

x + 2y + z = 6

2x + y + 2z = 6

x + y + z = 5

This system has

Solution:

(c )

∴ rank(A) = 2 ≠ 3 = rank() .

∴ The system is inconsistent and has no solution.

QUESTION: 5

Multiplication of matrices E and F is G. Matrices E and G are

What is the matrix F?

Solution:

(c)

Given

QUESTION: 6

Consider a non-homogeneous system of linear equations representing mathematically an over-determined system. Such a system will be

Solution:

Ans.(b)

In an over determined system having more equations than variables, it is necessary to have consistent having many solutions .

QUESTION: 7

For the set of equations

x_{1} + 2x + x_{3} + 4x_{4} = 0

3x_{1} + 6x_{2} + 3x_{3} + 12x_{4} = 0

Solution:

given set of equations are x_{1}+2x_{2}+x_{3}+4x_{4}=2 , 3x_{1}+6x_{2}+3x_{3}+12x_{4}=6

consider AB =

⇒ R_{2} → R_{2} - 3R_{1}

AB =

P(A)=1; P(AB)=1;n=4

⇒ P(A) =P(B) < no. of variables

⇒ Infinitely many solutions ⇒multiple non-trivial solution

QUESTION: 8

Let P ≠ 0 be a 3 × 3 real matrix. There exist linearly independent vectors x and y such that Px = 0 and Py = 0. The dimension of the range space of P is

Solution:

(b)

QUESTION: 9

The eigen values of a skew-symmetric matrix are

Solution:

(c)

QUESTION: 10

The rank of a 3×3 matrix C (=AB), found by multiplying a non-zero column matrix Aof size 3×1 and a non-zero row matrix B of size 1×3, is

Solution:

(b)

Let A =

Then C = AB =

Then det (AB) = 0.

Then also every minor

of order 2 is also zero.

∴ rank(C) =1.

QUESTION: 11

Match the items in columns I and II.

Column I Column II

P. Singular matrix 1. Determinant is not defined

Q. Non-square matrix 2. Determinant is always one

R. Real symmetric 3. Determinant is zero

S. Orthogonal matrix 4. Eigenvalues are always real

5. Eigenvalues are not defined

Solution:

(a) (P) Singular matrix → Determinant is zero

(Q) Non-square matrix → Determinant is not defined

(R) Real symmetric → Eigen values are always real

(S) Orthogonal → Determinant is always one

QUESTION: 12

Real matrices are given. Matrices [B] and

[E] are symmetric.

Following statements are made with respect to these matrices.

1. Matrix product [F]^{T} [C]^{T} [B] [C] [F] is a scalar.

2. Matrix product [D]^{T} [F] [D] is always symmetric.

With reference to above statements, which of the following applies?

Solution:

(a)

QUESTION: 13

The product of matrices (PQ)^{–1} P is

Solution:

(b)

(PQ)^{ -1} = P Q^{-1}P^{-1}P = Q^{-1}

QUESTION: 14

The matrix A

=

is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are

Solution:

Ans. (d)

QUESTION: 15

The inverse of the matrix is

Solution:

(b)

QUESTION: 16

The inverse of the 2 × 2 matrix is,

Solution:

(a).

QUESTION: 17

There are two containers, with one containing 4 Red and 3 Green balls and the other containing 3 Blue and 4 Green balls. One ball is drawn at random form each container.The probability that one of the ball is Red and the other is Blue will be

Solution:

QUESTION: 18

The Fourier transform of x(t) = e^{–at} u(–t), where u(t) is the unit step function,

Solution:

Ans. (d)

QUESTION: 19

Given that F(s) is the one-sided Laplace transform of f(t), the Laplace transform of is [EC:

Solution:

Ans. (b)

……(Lapalace formule)

QUESTION: 20

If f(t) is a finite and continuous function for t, the Laplace transformation is given by

For f(t) = cos h mt, the Laplace transformation is…..

Solution:

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