# Can you please help me with these questions, and I will appreciate it if you e
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Question
# Can you please help me with these questions, and I will appreciate it if you explain the reason for your answers in few words:
Question 1: Suppose A is an n x n matrix and det(A) = -84. Determine if the following statements are true or false.
a. A is invertible.
b. A is row equivalent to the identity matrix.
c. Ax = 0 has infinite solutions.
d. The transformation T(x) = Ax is one-to-one.
e. Ax = b has a unique solution for all b.
f. A is nonsingular.
Question 2: Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten axioms that fails.
a. The set of all quadratic functions whose graphs pass through the origin with the standard operations.
d. The set of all 2 x 2 diagonal matrices with the standard operations.
Explanation / Answer
1. a) True. A is invertible because det(A) is non zero.
b) True. Since both identity and A are invertible both are row equivalent.
c) False. Since A is invertible, Ax = 0 => A^(-1) A x = A^(-1) 0 => x = 0 is the only solution.
d) True. Ax is one-one, because if Ax = Ay => A^(-1) Ax = A^(-1) Ay => x = y.
e) True. For each b there is a unique solution of Ax = b i.e., x = A^(-1) b.
f) True.
2.a) False. The form of a quadratic function passing through origin is ax^2 + bx where a != 0. But sum of two quadratic functions may become linear (i.e., a = 0).
b) False. Let x > 0 in (x,y). But for scalar multiplication with -1, we have (-1) (x,y) = (-x, -y). Here, -x < 0.
c) True.
d) True.
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