Linear Algebra

Questions On Linear Algebra

Question 1

Consider the following 2 x 2 matrix

1	1
1	-1

Find the eigenvalues of the matrix A^19.

1. 1024 and -1024
2. 1024√2 and -1024√2
3. 4√2 and -4√2
4. 512√2 and -512√2

Answer: D

Explanation:

A = 

1	1
1	-1

A^2 = A x A = 

2	0
0	2

A^4 = A^2 x A^2 = 

4	0
0	4

A^8 = A^4 x A^4 = 

16	0
0	16

A^16 = A^8 x A^8 = 

256	0
0	256

A^19 = A^16 x A^2 x A^1 = 

512	512
512	-512

By applying characteristic polynomial:

512 - λ	512
512	-512 + λ

-(512 - λ) (512 + λ) - 512 * 512 = 0

λ^2 = 2 * 512^2

λ = 512√2 and -512√2

Question 2

Find the eigenvalues of the given matrix:

1	2	3
0	4	7
0	0	3

 

1. 1, 4, 3
2. 3, 7, 3
3. 7, 3, 2
4. 1, 2, 3

Answer: A

Explanation:

1 - λ	2	3
0	4 - λ	7
0	0	3 - λ

(1 - λ) * (4 - λ) * (3 - λ) = 0

λ = 1, 4, 3

Question 3

Consider the given matrix:

2	3
x	y

Given that the eigenvalues of the above matrix are 4 and 8, find the values of x and y.

1. x = 4, y = 10
2. x = 5, y = 8
3. x = -3, y = 9
4. x = -4, y = 10

Answer: D

Explanation:

2 - λ	3
x	y - λ

(2 - λ) * (y - λ) - 3x = 0

Given that eigenvalues are 4 and 8, we get:

(2 - 4) * (y - 4) - 3x = 0 and (2 - 8) * (y - 8) - 3x = 0

On solving the above equations, we get:

x = -4 and y = 10

Question 4

Find the determinant of the following matrix:

2	18	10
-4	-36	-20
7	63	35

1. 0
2. 1
3. 2
4. 3

Answer: A

Explanation: 

Observe that the second row can be made 0 by using first row by applying: 

R2 = R2 + 2*R1

The resultant matrix will be:

2	18	10
0	0	0
7	63	35

Since, the entire row is 0, its determinant will be 0.

Question 5

Which of the following statements is true?

1. If the determinant and the product of the trace of a matrix are positive, all its eigenvalues are positive.
2. If the trace of a matrix is positive, all its eigenvalues are positive.
3. If the determinant of a matrix is positive, all its eigenvalues are positive.
4. If the trace is positive and the determinant is negative, at least one eigenvalue is negative.

Answer: D

Explanation:

Trace is the sum of elements on the principal diagonal of the matrix. From the equation of eigenvalues, we get

1. The sum of the eigenvalues is equal to the trace of the matrix.
2. The product of the eigenvalues is equal to the determinant of the matrix.

Given that the trace is positive and the determinant is negative, there must be at least one eigenvalue with a negative value. More specifically, there can be an odd number of negative eigenvalues.

Question 6

Consider a set H of all 3 x 3 matrices of the following type:

a	f	e
0	b	d
0	0	c

where a,b,c,d,e,f can be any real numbers such that a*b*c != 0. The set H is

1. semigroup but not a monoid.
2. monoid but not a group
3. group
4. neither group nor semigroup

Answer: C

Explanation:

Given that a*b*c != 0, it means that it is a non-singular matrix, and its inverse is defined. So, the given set is a set of upper triangular matrices of size 3 x 3 that have a non-zero determinant. 

The product of two upper triangular matrices is also an upper triangular matrix. Thus, it forms an algebraic structure along with multiplication.

The set is associative since the multiplication of matrices follows the associative property. Therefore, it is a semigroup. The set is also a monoid since it contains an identity matrix. Since every matrix of H has an inverse, the set is a group.

Question 7

Consider the following equations having three unknowns:

2*x1 - x2 + 3*x3 = 1

3*x1 + 2*x2 + 5*x3 = 2

-x1 + 4*x2 + x3 = 3

The above system of equations has:

1. no solution
2. unique solution
3. more than one but finite number of solutions
4. infinite number of solutions

Answer: B

Explanation:

The matrix corresponding to the above equations will be:

2	-1	3
3	2	5
-1	4	1

The determinant of the matrix is:

= 2 * (2 - 20) + 1 * (3 + 5) + 3 * (12 + 2)

= 36 + 8 + 42

= 86

Question 8

Find the number of n x n symmetric matrices having each element either 0 or 1.

1. 2^n
2. 2^(n^2)
3. 2^((n^2 + n)/2)
4. 2^((n^2 - n)/2)

Answer: C

Explanation:

A 3 x 3 symmetric matrix will have the following properties:

a	b	c
d	e	f
g	h	i

b = d, c = g, f = h.

The upper half of a matrix contains n*(n+1)/2 elements. Once we fix the elements of this upper half, the lower half automatically gets fixed. The number of ways we can choose the upper half is if all elements are either 0 or 1 is:

2^(n*(n+1)/2) = 2^(n^2 + n)/2

Question 9

Find the eigenvalues of the following matrix:

2	-1
-4	5

1. -1, 1
2. 1, 6
3. 2, 5
4. 4, -1

Answer: B

Explanation: 

2 - λ	-1
-4	5 - λ

On solving the determinant of the above matrix, we will get:

λ^2 - 7λ + 6 = 0

The roots of this equation are 1 and 6.

Question 10

Let there be 4 n x n matrices - A, B, C, and D, each having a non-zero determinant. If ABCD = 1, then B^-1 is?

1. CDA
2. D^-1 C^-1 A^-1
3. ADC
4. None of the above

Answer: A

Explanation: 

Given:

ABCD = 1

Pre-Multiply both sides by A^-1:

BCD = A^-1

Pre-Multiply both sides by B^-1:

CD = B^-1A^-1

Post-Multiply both sides by A

CDA = B^-1

Conclusion

In this article, we discussed various quality multiple-choice questions based on the matrices and linear algebra. We hope that this blog has helped you enhance your knowledge of linear algebra and if you would like to learn more, check out our articles on code studio. Do upvote our blog to help other ninjas grow. Happy Coding!
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