The eigen values of a (2 × 2) matrix X are -2 and -3. The eigen values of the matrix (X + $$I$$) (X + 5$$I$$) are

Let the eigen values of a 2 × 2 matrix A be 1, -2 with eigen vectors x₁ and x₂ respectively. Then the eigen values and eigen vectors of the matrix A² - 3A + 4$$I$$ would, respectively, be

A

2, 14; x₁, x₂

B

2, 14; x₁ + x₂, x₁ - x₂

C

2, 0; x₁, x₂

D

2, 0; x₁ + x₂, x₁ - x₂

Consider the system of equations A_{(n × n)} X_{(n × 1)} = λ_{(n × 1)} where, λ is a scalar. Let (λ_i, x_i) be an eigen-pair of an eigen value and its corresponding eigen vector for real matrix A. Let $$I$$ be a(n × n) unit matrix. Which one of the following statement is NOT correct?

A

For a homogeneous n × n system of linear equations, (A - λ$$I$$)x = 0 having a nontrivial solution, the rank of (A - λ$$I$$) is less than n

B

For matrix A^m, m being a positive integer, (λ_i^m, x_i^m) will be the eigen-pair for all i

C

If A^T = A^-1, then |λ_i| = 1 for all i

D

If A^T = A, then λ_i is real for all i

Consider the following pattern of numbers: $\dfrac{\begin{matrix} 8 \\\ 6 \ \\\ 4 \ \end{matrix}}{ \ 6} \ \ \dfrac{\begin{matrix} 10 \\\ 5 \ \\\ 6 \ \end{matrix}}{ \ 11} \ \ \dfrac{\begin{matrix} 15 \\\ 7 \ \\\ 8 \ \end{matrix}}{ \ 16} \ \ \dfrac{\begin{matrix} 13 \\\ 4 \ \\\ 8 \ \end{matrix}}{ \ ?}$ What is the number at ? in the above pattern?

A

17

B

19

C

21

D

23

A 3 × 3 matrix has elements such that its trace is 11 and its determinant is 36. The eigen values of the matrix are all known to be positive integers. The largest eigen value of the matrix is

A

18

B

12

C

9

D

6

Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.

A

a, b, c

B

-a, -b, -c

C

a-b, b-a, c-a

D

a-1, b-1, c-1

The Eigen values of a 3×3 matrix are λ1, λ2, λ3 then the Eigen values of a matrix A3 are __________

A

λ1, λ2, λ3

B

$ \frac{1}{λ_1}, \frac{1}{λ_2}, \frac{1}{λ_3}$

C

$λ_1^3, λ_2^3, λ_3^3$

D

1, 1, 1

A real traceless 4 × 4 matrix has to eigen values -1 and +1. The other eigen values are

A

zero and +2

B

-1 and +1

C

zero and +1

D

+1 and +1

M is a 2 × 2 matrix with eigen values 4 and 9. The eigen values of M² are

A

-2 and -3

B

2 and 3

C

4 and 9

D

16 and 81

Consider the following statements regarding homogenous coordinate transformation matrix: 1. A homogenous transformation matrix can be considered to consist of four sub-matrices. 2. The upper left 3 × 3 sub-matrix represents the position vector. 3. The upper right 3 × 1 sub-matrix represents the rotation matrix. 4. The lower left 1 × 3 sub-matrix represents perspective transformation. Which of the above statements are correct?

A

1 and 3

B

1 and 4

C

2 and 3

D

2 and 4

The determinant of a 3 × 3 real symmetric matrix is 36. If two of its eigen values are 2 and 3 then the third eigen value is

A

4

B

6

C

8

D

9

The eigen values of a (2 × 2) matrix X are -2 and -3. The eigen values of the matrix (X + $$I$$) (X + 5$$I$$) are

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