Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1

# Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1

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## 20 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1

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Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 1

### The system of linear equations 4x + 2y = 7 2x + y = 6              has

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 1

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

Angemented matrix

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

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 2

### For the following set of simultaneous equations: 1.5x – 0.5y = 2 4x + 2y + 3z = 9 7x + y + 5z = 10

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 2

(a)

∴ rank of() = rank of(A) = 3

∴ The system has unique solution.

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 3

### The following set of equations has 3 x + 2 y + z = 4 x – y + z = 2 ​-2 x + 2 z = 5

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 3

(b)

∴ rank (A) = rank () = 3
∴ The system has unique solution

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 4

Consider the system of simultaneous equations
x + 2y + z = 6
2x + y + 2z =  6
x + y +  z = 5
This system has

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 4

(c )

∴ rank(A) = 2 ≠ 3 = rank() .

∴ The system is inconsistent and has no solution.

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 5

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

What is the matrix F?

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 5

(c)
Given

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 6

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 6

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

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 7

For the set of equations
x1 + 2x  + x3 + 4x4 = 0
3x1 + 6x2 + 3x3 + 12x4 = 0

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 7

given set of equations are x1+2x2+x3+4x4=2 , 3x1+6x2+3x3+12x4=6

consider AB =

⇒ R2 → R2 - 3R1

AB =

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

⇒ P(A) =P(B) < no. of variables

⇒ Infinitely many solutions ⇒multiple non-trivial solution

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - 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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 8

(b)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 9

The eigen values of a skew-symmetric matrix are

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 9

(c)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - 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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 10

(b)

Let A =

Then C = AB =

Then det (AB) = 0.

Then also every minor
of order 2 is also zero.
∴ rank(C) =1.

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 11

(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

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - 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?

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 12

(a)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 13

The product of matrices (PQ)–1 P is

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 13

(b)
(PQ) -1 = P Q-1P-1P = Q-1

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 14

Ans. (d)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 15

The inverse of the matrix     is

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 15

(b)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 16

The inverse of the 2 × 2 matrix    is,

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 16

(a).

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - 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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 17

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 18

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 18

Ans. (d)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 19

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Question 19

Ans. (b)

……(Lapalace formule)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - 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…..

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## GATE Mechanical (ME) 2023 Mock Test Series

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