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Problem1: a) If ۯ ൌ൥ͳ Ͳ ͳ ʹ െͳ ͳ ͳ͵ʹ ൩ ƒ†۰ ൌ ൥ ͳ Ͳ െ͵ െ͸ ͹ െ͵ െ͵ െͻ െʹ ൩, then find the values of Ǥ۽ ൌ ۷ݖ ൅ ۰ۯݕ ൅ ଶۯݔfSS…•ݖ†fݕ ǡݔ .a b) Find ݆ܽ݀ሺۯሻƒ†ۯିଵˆ‘Š‡ƒ‹š: 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 ={AM2×2(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 ൨M2×2(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={XM2×2(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 {XM2×2(Թ): X = XT} is a proper subspace of the vector space M2×2(Թ)?

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