Test: Eigenvalues & Eigenvectors - 1

# Test: Eigenvalues & Eigenvectors - 1

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Test: Eigenvalues & Eigenvectors - 1 - Question 1

### Find the sum of the Eigenvalues of the matrix Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 1

According to the property of the Eigenvalues, the sum of the Eigenvalues of a matrix is its trace that is the sum of the elements of the principal diagonal.
Therefore, the sum of the Eigenvalues = 3 + 4 + 1 = 8.

Test: Eigenvalues & Eigenvectors - 1 - Question 2

### Find the Eigenvalues of matrix Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 2

A - λI = 0 Now, After taking the determinant: (4 - λ)2 - 1 = 0

16 + λ2 - 8λ - 1 = 0

λ2 - 8λ + 15 = 0

- 3) (λ - 5) = 0

λ = 3, 5

Test: Eigenvalues & Eigenvectors - 1 - Question 3

### All the four entries of the 2 × 2 matrix are nonzero, and one of its eigen values is zero. Which of the following statements is true?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 3

One eigen value is zero Test: Eigenvalues & Eigenvectors - 1 - Question 4

The eigen values of the following matrix are Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 4

Let the matrix be A.  We know, Trace (A) = sum of eigen values. Test: Eigenvalues & Eigenvectors - 1 - Question 5

The three characteristic roots of the following matrix A Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 5 Test: Eigenvalues & Eigenvectors - 1 - Question 6

The sum of the eigenvalues of the matrix given below is Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 6 Test: Eigenvalues & Eigenvectors - 1 - Question 7

For which value of x will the matrix given below become singular? Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 7

Let the given matrix be A.  A is singular. Test: Eigenvalues & Eigenvectors - 1 - Question 8

Eigen values of a matrix are 5 and 1. What are the eigen values of the matrix S2  = SS?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 8

We know If λ be the eigen value of A ⇒λ2 is an eigen value of A2 .

For S matrix, if eigen values are λ1, λ2, λ3,... then for S² matrix, the eigen values will be λ²1 λ²2 λ²3......
For S matrix, if eigen values are 1 and 5 then for S² matrix, the eigen values are 1 and 25

Test: Eigenvalues & Eigenvectors - 1 - Question 9

The number of linearly independent eigenvectors of Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 9

Number of linear independent vectors is equal to the sum of Geometric Multiplicity of eigen values. Here only eigen value is 2.

To find Geometric multiplicity find n-r of (matrix-2I), where n is order and r is rank.

Rank of obtained matrix is 1 and n = 2 so n-r = 1. Therefore the no of linearly independent eigen vectors is 1

Test: Eigenvalues & Eigenvectors - 1 - Question 10

The eigenvectors of the matrix are written in the form . What is a + b?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 10     Test: Eigenvalues & Eigenvectors - 1 - Question 11

One of the Eigenvectors of the matrix A = is

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 11

The eigen vectors of A are given by  AX= λ X

So we can check by multiplication.   Test: Eigenvalues & Eigenvectors - 1 - Question 12

The minimum and the maximum eigen values of the matrix are –2 and 6, respectively. What  is the other eigen value?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 12 Test: Eigenvalues & Eigenvectors - 1 - Question 13

The state variable description of a linear autonomous system is, X= AX,

Where X is the two dimensional state vector and A is the system matrix given by The roots of the characteristic equation are

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 13

Characteristic equation will be :λ2 -4 =0 thus root of characteristic equation will be +2 and - 2.

Test: Eigenvalues & Eigenvectors - 1 - Question 14

For the matrix s one of the eigen values is equal to -2. Which of the following  is an eigen vector?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 14   Test: Eigenvalues & Eigenvectors - 1 - Question 15

x=[x1x2…..xn]T is an n-tuple nonzero vector. The n×n matrix V=xxT

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 15 Test: Eigenvalues & Eigenvectors - 1 - Question 16

Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix A9 equals

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 16  Test: Eigenvalues & Eigenvectors - 1 - Question 17

If the rank of a (5×6) matrix Q is 4, then which one of the following statements is correct?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 17

Rank of a matrix is equal to the No. of linearly independent row or no. of linearly  independent column vector.

Test: Eigenvalues & Eigenvectors - 1 - Question 18

The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues are

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 18   Test: Eigenvalues & Eigenvectors - 1 - Question 19

Identify which one of the following is an eigenvector of the matrix Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 19

Eigen Value (λ ) are 1,− 2. be the eigen  of A. Corresponding

To λ then.    be the eigen vector corrosponding to λ = 1

Test: Eigenvalues & Eigenvectors - 1 - Question 20

The eigenvalues of Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 20

The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix.

Hence 5, -19, and 37 are the eigenvalues of the matrix. Alternately, look atd λ = 5, -19, 37

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