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Eigen values, eigen vectors and diagonalization

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Beginner

Intermediate

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Find the determinant of the matrix

A - 2I

, where

A= \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 2 & 0 & 1 \end{bmatrix}

and

I

is the identity matrix of order

3 \times 3

.

a: 5 Explanation

Explanation

b: 3 Explanation

Explanation

c: -5 Explanation

Explanation

d: 8 Explanation

Explanation

Let

x^2 - x - 2 = 0

. Then x equals

a: 1, 2 Explanation

Explanation

b: -1, 2 Explanation

Explanation

c: 1, -2 Explanation

Explanation

d: -1, -2 Explanation

Explanation

Let

A = \begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix}

and

|A - aI| = 0

. Then a equals

a: 2, 1 Explanation

Explanation

b: 2, -1 Explanation

Explanation

c: -2, 1 Explanation

Explanation

d: -1, 1 Explanation

Explanation

Find

(x, y) \in \mathbb{R}^2

such that

2x + 0y = 0

.

a:

(0, y); y \in \mathbb{R}

Explanation

Explanation

b:

(0, x + y); x, y \in \mathbb{R}

Explanation

Explanation

c:

(x, y); x, y \in \mathbb{R}

Explanation

Explanation

d:

(x, 0); x \in \mathbb{R}

Explanation

Explanation

Let

A = \begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix}

. Then which of the following is a solution of

AX = 2X

.

a:

X = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

Explanation

Explanation

b:

X = \begin{bmatrix} 0 \\ 1 \end{bmatrix}

Explanation

Explanation

c:

X = \begin{bmatrix} 2 \\ -1 \end{bmatrix}

Explanation

Explanation

d:

X = \begin{bmatrix} 3 \\ 0 \end{bmatrix}

Explanation

Explanation

Let

T: \mathbb{R}^2 o \mathbb{R}^2

be a linear transformation defined as

T(x, y) = (x, 0)

, where

x, y \in \mathbb{R}

. Then

a:

(T + 2I)(x, y) = (x + 2, 0)

Explanation

Explanation

b:

(T + 2I)(x, y) = (3x, 2y)

Explanation

Explanation

c:

(T + 2I)(x, 2y) = (x - 2, 2y)

Explanation

Explanation

d:

(T + 2I)(3x, 2y) = (0, y + 2)

Explanation

Explanation

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