Linear Algebra - Linear Algebra

26:

Given Matrix A=

The eigen-values are

A.	(3,4,2)
B.	(3,4,1)
C.	(3,4,5)
D.	(2,3,5)
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	Option: B Explanation : Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

27:

Given: matrix A=

The eigen vector corresponding to the smallest eigen value is

A.	$\begin{bmatrix} 0\\0\\1 \end{bmatrix}$
B.	$\begin{bmatrix} 1\\0\\0 \end{bmatrix}$
C.	$\begin{bmatrix} 0\\1\\0 \end{bmatrix}$
D.	None of these
	Answer Report Discuss
	Option: A Explanation : For finding eigen vector corresponding to 1, putting λ equal to 1. $A-I=\begin{bmatrix} 2&0&0\\5&3&0\\3&6&0 \end{bmatrix}$ Rank of the coefficient matrix= = Number of non-zero diagonal elements = 2 < 3. Hence non-zero solution vector = 3 - 2 = 1 $\therefore \begin{bmatrix} 2&0&0\\5&3&0\\3&6&0 \end{bmatrix} = \begin{bmatrix} x_{1}\\x_{2}\\x_{3} \end{bmatrix}$ These equation can berewritten as $2x_{1}=0\\5x_{3}+2x_{2}=0\\3x_{3}+6x_{2}=0$ From these equations, $x_{1}=0 and x_{2}=0$ $x=\begin{bmatrix} x_{1}\\x_{2}\\x_{3} \end{bmatrix}=\begin{bmatrix} 0\\0\\x_{3} \end{bmatrix}=x_{3}\begin{bmatrix} 0\\0\\1 \end{bmatrix}$ Hence is the required eigen vector. $\begin{bmatrix} 0\\0\\1 \end{bmatrix}$ Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

28:

Rank the following (n+1)×(n+1)matrix, where a is areal number, is

A.	1
B.	2
C.	n
D.	dependent upon the value of A
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	Option: A Explanation : Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

29:

Matrices

commute under multiplication

A.	if a = b or θ = n $\pi$ , where n is an integer
B.	always
C.	never
D.	if a cosθ ≠ b sinθ
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	Option: A Explanation : Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

30:

Eigen values of the system represented by

are

A.	0, 0, 0, 0
B.	1, 1, 1, 1
C.	0, 0, 0, -1
D.	1, 0, 0, 0
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	Option: D Explanation : Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

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