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

2, 14; x1, x2

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

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

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

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

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 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 the matrix A be the idempotent matrix then the Eigen values of the idempotent matrix are ________

A

0, 1

B

0

C

1

D

-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

Consider a 3 × 3 real symmetric matrix S such that two of its eigen values are a ≠ 0, b≠ 0 with respective eigen vectors \[\left[ {\begin{array}{*{20}{c}} {{{\text{x}}_1}} \\ {{{\text{x}}_2}} \\ {{{\text{x}}_3}} \end{array}} \right

A

,\left.\

B

If a ≠ b then x₁y₁ + x₂y₂ + x₃y₃ equals

C

A. a

D

b

এক্সরশ্মির তরঙ্গ দৈর্ঘ্য 3̇A° হলে এর কম্পাংক কত?

A

1010 HZ

B

1015 HZ

C

1018 HZ

D

1021 HZ

λ এর কোন মানের জন্য 4î+2ĵ-3k^ এবং λî-3ĵ+2k^ ভেক্টরদ্বয় পরস্পর লম্ব হবে?

A

-3

B

3

C

12

D

-12

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

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