## Transcribed Text

Problem1:
a) If ۯ ൌͳ Ͳ ͳ
ʹ െͳ ͳ
ͳ͵ʹ
൩ ۰ ൌ ͳ Ͳ െ͵
െ െ͵
െ͵ െͻ െʹ
൩, then find the values of
Ǥ۽ ൌ ۷ݖ ۰ۯݕ ଶۯݔfSS
ݖfݕ ǡݔ .a
b) Find ݆ܽ݀ሺۯሻۯିଵ:
ۯ ൌͳ Ͳ ܽ
ʹ ܾ ܿ
െͳ ͳ ͳ
൩,
where ܾܽ ܾ ʹܽ െ ܿ ് Ͳ
c) Let A M3×3(Թ) with determinant ȁۯȁ = 2. Find ȁʹሺ݆ܽ݀ሺۯሻሻିଵ ۯȁ .
Problem 2:
a) Let A =
Ȃ
െ െ െ ૡ
െ
൩ and B =
Ȃ
ૡ
൩Ǥ Show that the matrices A and
B are row equivalent to each other.
b) Determine the value/s of D such that the following linear system:
x + 2y – z = 2
x – 2y + 3z = 1
x + 2y – (D2 – 3) z = D has: (i). no solution; (ii). unique solution; (iii). infinitely many solutions.
Problem 3:
a) Show that any homogeneous system of linear equations either has only the trivial solution or
infinitely many solutions and so every homogeneous linear system is consistent.
b) Give example of a homogeneous linear system with only the trivial solution.
c) Give example of a homogeneous linear system having infinitely many non-trivial solutions.
d) By using the Cramer’s rule, solve the following system:
x + 2y – z = 2
x + 3y + 3z = 2
x + 3y + 5z = 4.
Problem 4:
a) Let W ={AM2×2(Թ): AB = BA}, where B = ቂ
െ
ቃǤThen:
(i). Show that W is a vector subspace of the vector space M2×2(Թ).
(ii). Find a basis and dimension of W.
b) Find a basis of the vector space Թ͵ which contains the set {(1, 1, 0), (1, −1, 0)}.
Problem 5:
a) Show that A ={
ࢻ
J
ࢼ
G
൨M2×2(Թ): α+β = J - G}is a vector subspace of M2×2(Թ). Also find
a basis and dimension of the vector space A.
b) Show that S={XM2×2(Թ): X = - XT} is a subspace of the vector space M2×2(Թ) and
show further that the set {ቂ
െ
ቃ} is a basis for S.
c) Determine whether {XM2×2(Թ): X = XT} is a proper subspace of the vector space M2×2(Թ)?

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