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

2, 14; x1, x2

2, 14; x1 + x2, x1 - x2

2, 0; x1, x2

2, 0; x1 + x2, x1 - x2

Correct Answer: 2, 14; 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?

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

For matrix Am, m being a positive integer, (λim, xim) will be the eigen-pair for all i

If AT = A-1, then |λi| = 1 for all i

If AT = A, then λi 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, b, c

-a, -b, -c

a-b, b-a, c-a

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 __________

λ1, λ2, λ3

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

$λ_1^3, λ_2^3, λ_3^3$

1, 1, 1

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

0, 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

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

-3, -4

-1, -2

-1, -3

-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

,\left.\

If a ≠ b then x1y1 + x2y2 + x3y3 equals

A. a

b

Let A be the 2 × 2 matrix with elements a₁₁ = a₁₂ = a₂₁ = +1 and a₂₂ = -1. Then the eigen values of the matrix A¹⁹ are

1024 and -1024

1024√2 and -1024√2

4√2 and -4√2

512√2 and -512√2

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

zero and +2

-1 and +1

zero and +1

+1 and +1

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

-2 and -3

2 and 3

4 and 9

16 and 81