The matrix constructed by placing the Eigen vectors together is diagonalizing matrix.

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

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

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

City High School must put together a debating team consisting of four debaters.There are candidates of equal ability : X,Y and Z who are all seniors; and A,B,C and D who are all juniors..The school requires that there should be two seniors and two juniors in the team.It is also necessary that all of the debaters be able to work with one another. *Debaters Y and A cannot work together. *Debaters Z and C cannot work together. *Debaters A and B cannot work together. which of the following statements must be false? 1.Debaters Y and C are never selected together. 2.Debaters Z and D are never selected together. 3.Debaters Z and D are never selected together.

A

Only I

B

Only II

C

Only III

D

All of these

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 matrix constructed by placing the Eigen vectors together is diagonalizing matrix.

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