Prove that if A and B are 2x2 matrices, then (A + B)^(2) = A^(2)
+ AB + BA + B^(2). Hint: Write out 2x2 matrices for A and B using
variable entries in each location and perform the operations on
either side of the equation to determine whether the two sides are
equivalent.
Given is a sequence of the following pattern: {1, 2, 6, 24, 120,
720, …}
1. Write recursive equations for the above sequence.
2. Write a C++ recursive function that can compute the sequence
in 1. above of any
unsigned long number.
2. (a) Prove that U45 is generated by the set {14,28}.
(b) Prove that the additive group Z×Z is generated by the set
S={(3,1),(−2,−1),(4,3)}.
Please be thorough step by step with details, please.
(1)Prove that for every a, b ∈ R, |a + b| = |a| + |b| ⇐⇒ ab ≥ 0.
Hint: Write |a + b| 2 = (|a| + |b|) 2 and expand.
(2) Prove that for every x, y, z ∈ R, |x − z| = |x − y| + |y −
z| ⇐⇒ (x ≤ y ≤ z or z ≤ y ≤ x). Hint: Use part (1) to prove part
(2).
(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then
a∣c.
(B) Let p ≥ 2. Prove that if 2p−1 is prime, then p
must also be prime.
(Abstract Algebra)