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Linear Algebra MCQ Level - 2 - Question 1

The system of the equations,

*x* – 4*y* + 7*z* = 14

3*x* + 8*y* – 2*z* = 13

7*x* – 8*y* + 26*z* = 5

is :

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 1

Then, consider the augmented matrix,

∵ Rank ** A** ≠ Rank [

∴ The system is inconsistent.

The correct answer is: Inconsistent

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 2

**tr( A) = 0, |A| = 1**

∴ The characteristic polynomial will be,

t(t) = t

Now, ** A** is a matrix over the field .Then

So, ** A** is not diagonalisable over .

Now, if

*t*(*t*) = *t*^{2} – 1

has two roots ** i** and

Thus, ** A** has two distinct eigenvalues

Hence ** A** has two independent eigenvectors. Accordingly,

The correct answer is: * A* is diagonalisable over

Linear Algebra MCQ Level - 2 - Question 3

For what values of ** k**, the given system of equations will have a unique solution?

2

x + 2(k + 1)y + (3k + 4)z = 0

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 3

For the unique solution. **| A|** should be non-zero where

⇒ 2*k*^{2} + 8 ≠ 0

⇒ *k*^{2} ≠ 4

⇒ *k* ≠ ± 2

The correct answer is: *k* ≠ ±2

Linear Algebra MCQ Level - 2 - Question 4

Which of the following will satisfy the given system?

2*x*_{1} – 3*x*_{2} – 6*x*_{3} – 5*x*_{4} + 2*x*_{5} = 7

*x*_{3} + 3*x*_{4} – 7*x*_{5} = 6

x_{4} – 2x_{5} = 1

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 4

Consider the augmented matrix,

We can see that the pivot elements exists in columns 1, 3 and 4

So, *x*_{2} and *x*_{5} are free variables.

Given, 2*x*_{1} – 3*x*_{2} – 6*x*_{3} – 5*x*_{4} + 2*x*_{5} = 7 ...(1)

*x*_{3} + 3*x*_{4} – 7*x*_{5} = 6 ...(2)

*x*_{4} – 2*x*_{5} = 1 ...(3)

from (3),

*x*_{4} = 1 + 2*x*_{5}

from (2),

*x*_{3} + 3(1 + 2*x*_{5}) – 7*x*_{5} = 6

*x*_{3} = 3 + *x*_{5}

from (1),

2*x*_{1} – 3*x*_{2} –6(3 + *x*_{5}) –5(1 + 2*x _{5}*) + 2

⇒ 2

⇒ 2

⇒ 2

∵ *x*_{2} and *x*_{5} are free variables, Let *x*_{2} = *x*_{5} = 0

then (15, 0, 3, 1, 0)* ^{T}* becomes one of the solutions.

The correct answer is: (15, 0, 3, 1, 0)

Linear Algebra MCQ Level - 2 - Question 5

The system of equations,

*x* + *y* + *z* = 0

2*x* – *y* – 3*z* = 0

3*x* – 5*y* + 4*z* = 0

*x* + 17*y* + 4*z* = 0

Will have :

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 5

Consider the coefficient matrix,

Rank of the coefficient matrix is 3 which is equal to the number of unknowns.

The given system of equations possesses non zero solution.

The correct answer is: Unique solution

Linear Algebra MCQ Level - 2 - Question 6

Choose the correct option about the system,

3*x* + *y* + *z* = 8

–*x* + *y* – 2*z* = –5

*x* + *y* + *z* = 6

–2*x* + 2*y* – 3*z* = –7

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 6

Consider the augmented matrix,

∴ Rank of coefficient matrix = Rank of augmented matrix = Number of unknowns

Hence, unique solution, the system reduces to,

–3*z* = – 9 ⇒ *z* = 3

2*y* – *z* = 1 ⇒ *y* = 2

*x + y + z* = 6 ⇒ *x* = 1

∴ The solution is (1, 2, 3)* ^{T}*.

The correct answer is: The system is consistent having unique solution, (1, 2, 3)^{T}

Linear Algebra MCQ Level - 2 - Question 7

A necessary and sufficient condition that values, not all zero may be assigned to n variables x_{1}, x_{2}, ......., x_{n} so that the homogeneous equations, a_{i1}x_{1} + a_{i2}x_{2} + ..... + a_{in}x_{n} = 0 (i = 1, 2,.....n) hold simultaneously, is :

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 7

For the scalars, a_{i1}, a_{i2}, ..........., a_{in} (i,= 1, 2,.....,n)

not all zero such that a_{i1}x_{1} + a_{i2}x_{2} + ...... + a_{in} x_{n} = 0

⇔ that {x_{1}, x_{2},..........,x_{n}} is linearly dependent.

⇒ Rank of coefficient matrix will be less than. Hence, minor of order n in the matrix or determinant of the coefficient matrix will be zero.

The correct answer is: |a_{ij}|_{n × n} = 0

Linear Algebra MCQ Level - 2 - Question 8

Consider the following statements about the given system of equations,

*x* + 2*y* + 3*z* = 0

3*x* + 4*y* + 4*z* = 0

7*x* + 10*y* + 12*z* = 0

and choose the correct one

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 8

Coefficient matrix to the given system of equation is,

Hence, the rank of the coefficient matrix is 3 which is equal to the number of unknown variables in the system.

∴ The system has a unique solution.

The correct answer is: The given system is consistent and has a unique solution.

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 9

Since, *M* is upper triangular, *M*^{2} will also be upper triangular and hence, both of them will be diagonalisable.

The correct answer is: Both *M* and *M*^{2} are diagonalisable

Detailed Solution for Linear Algebra MCQ Level - 2 - Question 10

Determinant A = cos^{2} θ +sin^{2} θ = 1

Hence, A is non-singular and A^{-1}exists

The correct answer is: A is non singular and A^{–1}^{ }exists

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