Question 1 |

A | |

B | |

C | |

D |

**GATE CS 2013**

**Linear Algebra**

**Discuss it**

First of all, you should know the basic properties of determinants before approaching For these kind of problems. 1) Applying any row or column transformation does not change the determinant 2) If you interchange any two rows, sign of the determinant will change

A = | 1 x x^2 | | 1 y y^2 | | 1 z z^2 |

To prove option (b)

=> Apply column transformation C2 -> C2+C1

C3 -> C3+C1

=> det(A) = | 1 x+1 x^2+1 | | 1 y+1 y^2+1 | | 1 z+1 z^2+1 |

To prove option (c),

=> Apply row transformations R1 -> R1-R2

R2 -> R2-R3

=> det(A) = | 0 x-y x^2-y^2 | | 0 y-z y^2-z^2 | | 1 z z^2 |

To prove option (d),

=> Apply row transformations R1 -> R1+R2

R2 -> R2+R3

=> det(A) = | 2 x+y x^2+y^2 | | 2 y+z y^2+z^2 | | 1 z z^2 |

This solution is contributed by **Anil Saikrishna Devarasetty** .

Question 2 |

^{19}are

A | |

B | |

C | |

D |

**GATE CS 2012**

**Linear Algebra**

**Discuss it**

A = 1 1 1 -1 A^{2}= 2 0 0 2 A^{4}= A^{2}X A^{2}A^{4}= 4 0 0 4 A^{8}= 16 0 0 16 A^{16}= 256 0 0 256 A^{18}= A^{16}X A^{2}A^{18}= 512 0 0 512 A^{19}= 512 512 512 -512 Applying Characteristic polynomial 512-lamda 512 512 -(512+lamda) = 0 -(512-lamda)(512+lamda) - 512 x 512 = 0 lamda^{2 = 2 x 5122 }

Question 3 |

1, 4, 3 | |

3, 7, 3 | |

7, 3, 2 | |

1, 2, 3 |

**GATE CS 2011**

**Linear Algebra**

**Discuss it**

The Eigen values of a triangular matrix are given by its diagonal entries. We can also calculate (or verify given answers) using characteristic equation obtained by |M - λI| = 0.

1-λ 2 3

0 4-λ 7 = 0

0 0 3-λ

Which means

(1-λ)(4-λ)(3-λ) = 0

Question 6 |

0 |

**Linear Algebra**

**GATE-CS-2014-(Set-2)**

**Discuss it**

Question 7 |

1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 1 0 0 0 1is ______

4 | |

5 | |

6 | |

7 |

**Linear Algebra**

**GATE-CS-2014-(Set-2)**

**Discuss it**

Question 8 |

If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative. | |

If the trace of the matrix is positive, all its eigenvalues are positive.
| |

If the determinant of the matrix is positive, all its eigenvalues are positive. | |

If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive. |

**Linear Algebra**

**GATE-CS-2014-(Set-3)**

**Discuss it**

Question 9 |

a group | |

a monoid but not a group | |

a semigroup but not a monoid | |

neither a group nor a semigroup |

**Linear Algebra**

**GATE-CS-2005**

**Discuss it**

Question 10 |

no solution | |

a unique solution | |

more than one but a finite number of solutions | |

an infinite number of solutions |

**Linear Algebra**

**GATE-CS-2005**

**Discuss it**

2 -1 3 3 -2 5 -1 4 1

Question 11 |

-1 and 1 | |

1 and 6 | |

2 and 5 | |

4 and -1 |

**Linear Algebra**

**GATE-CS-2005**

**Discuss it**

2-λ -1 -4 5-λWe get the equation as λ

^{2}-7 λ + 6 = 0 which gives us eigenvalues as 6 and 1.

Question 12 |

^{-1}is

D ^{-1}C^{-1}A^{-1} | |

CDA | |

ADC | |

Does not necessarily exist |

**Linear Algebra**

**GATE-CS-2004**

**Discuss it**

Question 13 |

-x + 5y = -1 x - y = 2 x + 3y = 3

infinitely many | |

two distinct solutions | |

unique | |

none of these |

**Linear Algebra**

**GATE-CS-2004**

**Discuss it**

Question 14 |

≤ a + b | |

≤ max {a, b} | |

≤ min {M-a, N-b} | |

≤ min {a, b} |

**Linear Algebra**

**GATE-CS-2004**

**Discuss it**

**(D)**is correct. Source: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2004.html

Question 15 |

0 | |

1 | |

2 | |

infinitely many |

**Linear Algebra**

**GATE-CS-2003**

**Discuss it**

Question 16 |

4 | |

2 | |

1 | |

0 |

**Linear Algebra**

**GATE-CS-2002**

**Discuss it**

Rank of the matrix is defined as the maximum number of linearly independent vectors (or) the number of non-zero rows in its row-echelon matrix. A = | 1 1| | 0 0| Since, the matrix A is already in echelon form, Just count the number of non-zero rows to get the rank of the matrix = 1.

Please refer http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29 This solution is contributed by**Anil Saikrishna Devarasetty**.

Question 17 |

S1: The sum of two singular n × n matrices may be non-singular S2: The sum of two n × n non-singular matrices may be singular.Which of the following statements is correct?

S1 and S2 are both true | |

S1 is true, S2 is false | |

S1 is false, S2 is true | |

S1 and S2 are both false |

**Linear Algebra**

**GATE-CS-2001**

**Discuss it**

**S1 is True: The sum of two singular n × n matrices may be non-singular**It can be seen be taking following example. The following two matrices are singular, but their sum is non-singular.

M1 and M2 are singular M1 = 1 1 1 1 M2 = 1 -1 -1 1 But M1+M2 is non-singular M1+M2 = 2 0 0 2

**S2 is True: The sum of two n × n non-singular matrices may be singular**

M1 and M2 are non-singular M1 = 1 0 0 1 M2 = -1 0 0 -1 But M1+M2 is singular M1+M2 = 0 0 0 0

Question 18 |

5 | |

0 | |

4 | |

20 |

**Linear Algebra**

**GATE-CS-2000**

**Discuss it**

Question 19 |

| 2 2 | | 4 9 |, if the diagonal elements of U are both 1, then the lower diagonal entry l

_{22}of L is

4 | |

5 | |

6 | |

7 |

**Linear Algebra**

**GATE-CS-2015 (Set 1)**

**Discuss it**

| 2 2 | = | l_{11 }0 | * | 1 u_{12}| | 4 9 | | l_{21}l_{22}| | 0 1 | l_{21}* u_{12}+ l_{22}* 1 = 9 ------ (1) We need to find l_{21}and u_{12}l_{21}* 1 + l_{22}* 0 = 4ll_{21}= 4_{11}* U_{12}+ 0 * 1 = 2 l_{11}= 2UPutting value of l_{12}= 1_{21}and u_{12}in (1), we get 4 * 1 + l_{22}* 1 = 9 l_{22}= 5

Question 20 |

A = | 1 4 | | b a |

a = 6, b = 4 | |

a = 4, b = 6 | |

a = 3, b = 5 | |

a = 5, b = 3 |

**Linear Algebra**

**GATE-CS-2015 (Set 1)**

**Discuss it**

The character equation for given matrix is | 1-λ 4 | = 0 | b a-λ| (1-λ)*(a-λ) - 4b = 0 Putting λ = -1, => (1 - (-1)) * (a - (-1)) - 4b = 0 => 2a + 2 - 4b = 0 => 2b - a = 1 Putting λ = 7, => (1 - 7) * (a - 7) - 4b = 0 => -6a + 42 - 4b = 0 => 2b + 3a = 21 Solving the above two equations, we get a = 5, b = 3

Question 21 |

⎡ 4 5 ⎤ ⎣ 2 1 ⎦is ____

5 | |

6 | |

7 | |

8 |

**Linear Algebra**

**GATE-CS-2015 (Set 2)**

**Discuss it**

The character equation for given matrix is | 4-λ 5 | = 0 | 2 1-λ| (4-λ)*(1-λ) - 10 = 0 λ^{2}- 5λ - 6 = 0 (λ+1)*(λ-6) = 0 λ = -1, 6 Greater of two Eigenvalues is 6.

Question 22 |

⎡ 3 4 45⎤ ⎢ 7 8 105⎥ ⎣13 2 195⎦ 1. Add the third row to the second row. 2. Subtract the third column from the first column.The determinant of the resultant matrix is _____________.

0 | |

1 | |

50 | |

100 |

**Linear Algebra**

**GATE-CS-2015 (Set 2)**

**Discuss it**

Question 23 |

⎡ 1 -1 2 ⎤ ⎢ 0 1 0 ⎥ ⎣ 1 2 1 ⎦

A | |

B | |

C | |

D |

**Linear Algebra**

**GATE-CS-2015 (Set 3)**

**Discuss it**

And let the given matrix be A (square matrix of order 3 x3) The characteristic equation for this is : AX = zX ( X is the required eigenvector ) AX - zX = 0 [ A - z I ] [X] = 0 ( I is an identity matrix of order 3 ) put z = 1 ( because one of the eigenvalue is 1 ) [ A - 1 I ] [X] = 0 The resultant matrix is : [ 0 -1 2 ] [x1] [0] | 0 0 0 ] |x2] =|0| [ 1 2 0 ] |x3] [0] Multiplying thr above matrices and getting the equations as: -x2 + 2x3 = 0 ----------------(1) x1 + 2x2 = 0-----------------(2) now let x1 = k, then x2 and x3 will be -k/2 and -k/4 respectively. hence eigenvector X = { (k , -k/2, -k/4) } where k != 0 put k = -4c ( c is also a constant, not equal to zero ), we get X = { ( -4c, 2c, 1c ) }, i.e. { c ( -4, 2, 1 ) } Hence option B.

Question 24 |

px + qy + rz = 0 qx + ry + pz = 0 rx + py + qz = 0then which one of the following options is True?

p – q + r = 0 or p = q = –r | |

p + q – r = 0 or p = –q = r | |

p + q + r = 0 or p = q = r | |

p – q + r = 0 or p = –q = –r |

**Linear Algebra**

**GATE-CS-2015 (Set 3)**

**Discuss it**

**Nitika Bansal.**

Question 25 |

(∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)] | |

(∃x)(∀y)[(a(x, y) ∨ b(x, y)) ∧¬ c(x, y)] | |

¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)] | |

¬ (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)] |

**Linear Algebra**

**GATE-IT-2004**

**Discuss it**

Question 26 |

x=6,y=3,z=2 | |

x=12,y=3,z=—4 | |

x=6,y=6,z=—4 | |

x=12,y=—3,z=O |

**Linear Algebra**

**GATE-IT-2004**

**Discuss it**

Question 27 |

*H1,H2,H3,...*be harmonic numbers Then, for can be expressed as

nHn+1 - (n + 1) | |

(n + 1)Hn - n | |

nHn - n | |

(n+1)Hn+1—(n+1) |

**Linear Algebra**

**GATE-IT-2004**

**Discuss it**

Question 28 |

0 | |

1 | |

15 | |

-1 |

**Linear Algebra**

**GATE-CS-2016 (Set 1)**

**Discuss it**

^{2})*(3) = (5)*(3) = 15. Thus, C is the required answer.

Question 29 |

8 | |

7 | |

9 | |

10 |

**Linear Algebra**

**GATE-CS-2016 (Set 2)**

**Discuss it**

Question 30 |

I. If m < n, then all such systems have a solution II. If m > n, then none of these systems has a solution III. If m = n, then there exists a system which has a solutionWhich one of the following is CORRECT?

I, II and III are true | |

Only II and III are true | |

Only III is true | |

None of them is true |

**Linear Algebra**

**GATE-CS-2016 (Set 2)**

**Discuss it**

**Mohit Gupta**.

Question 31 |

^{−1})

^{T}is _________ [This Question was originally a Fill-in-the-blanks Question]

1/8 | |

1 | |

1/4 | |

2 |

**Linear Algebra**

**GATE-CS-2016 (Set 2)**

**Discuss it**

Question 32 |

2 ^{n}.3^{n(n-1)/2} | |

2 ^{n} | |

n ^{2} | |

n |

**Linear Algebra**

**GATE 2017 Mock**

**Discuss it**

By Product rule,

Number of antisymmetric relations possible on A = 2^n. 3^(n(n-1)/2).