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QUESTION: 1

Given the matrix the eigenvector is

Solution:

Characteristic equation

QUESTION: 2

The eigen values and the corresponding eigen vectors of a 2 × 2 matrix are given by

Solution:

We know, sum of eigen values = trace (A). = Sum of diagonal element of A.

Therefore λ1 + λ 2 = 8 + 4 =12

Option (a)gives, trace(A) = 6 + 6 =12.

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?

Solution:

One eigen value is zero

QUESTION: 4

The eigen values of the following matrix are

Solution:

Let the matrix be A. We know, Trace (A)=sum of eigen values.

QUESTION: 5

The three characteristic roots of the following matrix A

Solution:

A is lower triangular matrix. So eigen values are only the diagonal elements.

QUESTION: 6

The sum of the eigenvalues of the matrix given below is

Solution:

Sum of eigen values of A= trace (A)

QUESTION: 7

For which value of x will the matrix given below become singular?

Solution:

Let the given matrix be A. A is singular.

QUESTION: 8

Eigen values of a matrix are 5 and 1. What are the eigen values of the matrix S^{2} = SS?

Solution:

We know If λ be the eigen value of A ⇒λ^{2} is an eigen value of A^{2} .

QUESTION: 9

The number of linearly independent eigenvectors of

Solution:

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

QUESTION: 10

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

Solution:

QUESTION: 11

One of the Eigenvectors of the matrix A = is

Solution:

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

So we can check by multiplication.

QUESTION: 12

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

Solution:

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

Solution:

Characteristic equation will be :λ^{2} -4 =0 thus root of characteristic equation will be +2 and - 2.

QUESTION: 14

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

Solution:

QUESTION: 15

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

Solution:

As every minor of order 2 is zero.

QUESTION: 16

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

A^{9} equals

Solution:

QUESTION: 17

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

Solution:

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

QUESTION: 18

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

Solution:

QUESTION: 19

Identify which one of the following is an eigenvector of the matrix

Solution:

Eigen Value (λ ) are 1,− 2.

be the eigen of A. Corresponding

To λ then.

be the eigen vector corrosponding to λ = 1

QUESTION: 20

The eigenvalues of

Solution:

**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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