M132: LINEAR ALGEBRA
Student Component
Student Name: ABDULLAH SAUD ALHAZMI
Student ID Number: 50902019
Course Section Number: M132
Tutor Component
QUESTION 1 2 3 4 5 6
MARK 10 10 10 10 10 10
SCORE
TOTAL
Tutor’s Comments:
Q−1:[5×2 marks]
Answer each of the following as True or False (justify your answer):
If the matrices A and B commute, thenA2B = BA2.
True. A2B = AAB = ABA=BAA=BA2
The reduced row echelon form of the matrix is . True
[■(3&3&1@3&-1&0@-1&-1&2)] □(→┴(swap R_3 by R_1 then-R_1 ) ) [■(1&1&-2@3&-1&0@3&3&1)] □(→┴(R_2-3R_1&R_3-3R_1 ) )
[■(1&1&-2@0&-4&6@0&0&7)] □(→┴(-1/4 R_2 ) ) [■(1&1&-2@0&1&2⁄3@0&0&7)] →┴(R_1-R_2 ) [■(1&0&(-8)⁄3@0&1&2⁄3@0&0&7)] →┴( 1/7 R_2 ) [■(1&0&(-8)⁄3@0&1&2⁄3@0&0&1)]
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True
If A is invertible and has inverse A^(-1)=B so AB=I and BA=I as AA^(-1)=I
A^2-3A+I=0 let I=AA^(-1)
A^2-3A+AA^(-1)=0
-AA^(-1)=A^2-3A
A^(-1)=-(A^2-3A)/A=-(A-3I)=(3I-A)
Or
A(A-3)=-I so A^(-1)=3-A
d)The linear system having the coefficient matrix is inconsistent if ʎ = 6 or ʎ = 4. True if ℷ=6 coff matrix=[■(0&0&0@0&1&-1@0&0&2)]
As row &column of coff matrix consists entirely of zeros, then |A| = 0. So system is inconsistent. if ℷ=4 coff matrix=[■(-2&0&0@0&1&-1@0&0&0)]
As row of coff matrix consists entirely of zeros, then |A| = 0. So system is inconsistent.
e) Let . The set is linearly independent.
True
Assume that:c_1 v_1+c_2 v_2=0 c_1 [■(1@1)]+c_2 [■(3@-1)] =[■(0@0)]
[├ ■(c_1&〖3c〗_2@c_1&-c_2 )┤| ■(0@0)]
[├ ■(1&3@1&-1)┤| ■(0@0)] →┴(R_2-R_(1 ) ) [├ ■(1&3@0&-4)┤| ■(0@0)] →┴(-1/4 R_2 ) [├ ■(1&3@0&1)┤| ■(0@0)] →┴(R_1-〖3R〗_(2 ) ) [├ ■(1&0@0&1)┤| ■(0@0)]
From elimination original matrix to get values of c_1,c_2we find that they have many values depend on value of c_4 hence c_1,c_2,c_3,c_4 equal 0 so S={v1,v2,}is linearly independent
Q−2: [4+3+3