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

Correct Answer: $λ_1^3, λ_2^3, λ_3^3$

If λ1, λ2, λ3……… λn are the Eigen values of matrix A then the Eigen values of matrix Am are said to be $λ_1^m, λ_2^m, λ_3^m,………λ_n^m$. So, the answer is $λ_1^3, λ_2^3, λ_3^3$.

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

2, 14; x1 + x2, x1 - x2

2, 0; x1, x2

2, 0; x1 + x2, x1 - x2

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

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

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

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

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

0, 1

-1

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

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