Eigen values of a 4 × 4 matrix [A] are given as 2, -3, 13 and 7. Then, what is the value of |det[A]| ?

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

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

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

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

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

A

-3, -4

B

-1, -2

C

-1, -3

D

-2, -4

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

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

Eigen values of a 4 × 4 matrix [A] are given as 2, -3, 13 and 7. Then, what is the value of |det[A]| ?

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