Q1. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, = (0,1,2) and x, =(-3,7,2) are linearly dependentand find the relation between them.Ans: 9x, - 12x, + 5х, - 5х, -0Q2. If the vectors (0,1,a), (1, a,1) and (a,1,0) is linearly dependent, then find the value of a.Ans: 0,+/2Q3. Find the eigen values and eigen vectors of the following matrices:8 -6 2(i) -6 7[31 4](ii) 0 2 60 o 5-4-43Ans: (i) 0, 3, 15, k 2, ka(ii) 3, 2, 5,Q4. Verify Cayley-Hamilton theorem for the following matrix and hence compute A:[2 -1 1]A = -1 2 -1I -1 2[3 1 -1Ans: 41 3 13[2 11Q5. Find the characteristic equation of the matrix A =0 1 0 and hence, compute A. Also find thematrix represented by A -5A" +7A“ - 3A +A* - SA' + 8A? - 2A +1.[8 5 5[ 2-1 -1Ans: 2'-5a + 72 - 3 = 0, 0 3 0,A355 83-1 -1105Q6. Show that the matrix -2 -3 -4 has less than three linearly independent eigen vectors. Also357find them.Ans: A= 2,2,3. For i = 3, X, = [k,k,-2k] , for 2 = 2, X, = [5k,2k,-Sk][i -12Q7. Reduce the matrix A = 0 2 -1 to diagonal form by similarity transformation. Hence find A'.jo 0 31 -7 32Ans: 0 8 - 190 027Q8..Test the consistency and hence solve the following set of equations: x, + 2x, + x, =2,Зх, +х, - 2х, -1, 4x, — 3х, -х, -3, 2х, +4х, +2х, -4.Ans: x, =1, x, =0 x, =19.3-9-13-7.-1Q9 Obtain the Model matrix A=I-74Ans. M = 1 0 1L-1 -1 -3]Q10. Find a matrix P which diagonalizes the matrix A=: . Varify P-AP = D where Dis the diagonal Matrix.Ans. P=-2[2 07D=0 5

Answered: Image /qna-images/answer/5a917148-894f-466f-8f5d-b5ca9f0e43ff.jpg

Answered: QI. Show that the vectors x, =(1,2,4),…

QI. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, (0,1,2) and x, =(-3,7,2) are linearly dependent and find the relation between them.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter1: Vectors

Section1.3: Lines And Planes

Problem 18EQ

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Question

100%

Q1. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, = (0,1,2) and x, =(-3,7,2) are linearly dependent
and find the relation between them.
Ans: 9x, - 12x, + 5х, - 5х, -0
Q2. If the vectors (0,1,a), (1, a,1) and (a,1,0) is linearly dependent, then find the value of a.
Ans: 0,+/2
Q3. Find the eigen values and eigen vectors of the following matrices:
8 -6 2
(i) -6 7
[31 4]
(ii) 0 2 6
0 o 5
-4
-4
3
Ans: (i) 0, 3, 15, k 2, ka
(ii) 3, 2, 5,
Q4. Verify Cayley-Hamilton theorem for the following matrix and hence compute A:
[2 -1 1]
A = -1 2 -1
I -1 2
[3 1 -1
Ans: 41 3 1
3
[2 11
Q5. Find the characteristic equation of the matrix A =0 1 0 and hence, compute A. Also find the
matrix represented by A -5A" +7A“ - 3A +A* - SA' + 8A? - 2A +1.
[8 5 5
[ 2
-1 -1
Ans: 2'-5a + 72 - 3 = 0, 0 3 0,A
3
55 8
3
-1 -1
10
5
Q6. Show that the matrix -2 -3 -4 has less than three linearly independent eigen vectors. Also
3
5
7
find them.
Ans: A= 2,2,3. For i = 3, X, = [k,k,-2k] , for 2 = 2, X, = [5k,2k,-Sk]
[i -1
2
Q7. Reduce the matrix A = 0 2 -1 to diagonal form by similarity transformation. Hence find A'.
jo 0 3
1 -7 32
Ans: 0 8 - 19
0 0
27
Q8..Test the consistency and hence solve the following set of equations: x, + 2x, + x, =2,
Зх, +х, - 2х, -1, 4x, — 3х, -х, -3, 2х, +4х, +2х, -4.
Ans: x, =1, x, =0 x, =1
9.
3
-9
-1
3
-7.
-1
Q9 Obtain the Model matrix A=
I-7
4
Ans. M = 1 0 1
L-1 -1 -3]
Q10. Find a matrix P which diagonalizes the matrix A
=: . Varify P-AP = D where D
is the diagonal Matrix.
Ans. P=
-2
[2 07
D=
0 5

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