answer:False. Using the result from the original proof, we have that A^{4n+2} = (A^4)^n cdot A^2 = I cdot A^2 = A^2 = begin{pmatrix} -1&0&0 0&1&0 0&0&-1 end{pmatrix} which is not equal to the identity matrix.

answer:Yes, the configuration shown in the original answer is unique up to Euclidean motions. Any other configuration would have to contain a diamond shape with four points, and the remaining three points would have to be placed at the vertices of an equilateral triangle centered at the center of the diamond. This is the only way to ensure that the distance between any two points in the set is 1.

answer:To factor 71955 into its prime factors, we can start by finding the smallest prime number that divides it. We can see that 3 is the smallest prime number that divides 71955, so we can write: 71955 = 3 cdot 23985 Now, we can repeat this process with 23985. We can see that 5 is the smallest prime number that divides 23985, so we can write: 23985 = 5 cdot 4797 We can continue this process until we reach a point where the remaining number is prime. In this case, we find that 4797 is prime, so we can write: 4797 = 4797 Therefore, the prime factorization of 71955 is: 71955 = 3^3cdot 5^1cdot 13^1cdot 41^1 The answer is 3^3cdot 5^1cdot 13^1cdot 41^1

answer:1. Find the first derivative of the function: f'(x) = frac{d}{dx} sqrt{1-(5 x-1)^2} = frac{1}{2sqrt{1-(5 x-1)^2}} cdot frac{d}{dx} (1-(5 x-1)^2) = frac{1}{2sqrt{1-(5 x-1)^2}} cdot (-2(5 x-1)) cdot 5 = -frac{5(5 x-1)}{sqrt{1-(5 x-1)^2}} 2. Find the second derivative of the function: f''(x) = frac{d}{dx} f'(x) = frac{d}{dx} left(-frac{5(5 x-1)}{sqrt{1-(5 x-1)^2}}right) = -frac{5}{sqrt{1-(5 x-1)^2}} cdot frac{d}{dx} (5 x-1) = -frac{5}{sqrt{1-(5 x-1)^2}} cdot 5 = -frac{25}{sqrt{1-(5 x-1)^2}} 3. Simplify the expression: f''(x) = -frac{25}{sqrt{1-(5 x-1)^2}} = -frac{25}{(1-(5 x-1)^2)^{1/2}} = -frac{25}{(1-(5 x-1)^2)^{3/2}} cdot frac{(1-(5 x-1)^2)^{1/2}}{(1-(5 x-1)^2)^{1/2}} = -frac{25}{(1-(5 x-1)^2)^{3/2}} cdot frac{1}{sqrt{1-(5 x-1)^2}} = -frac{25}{(1-(5 x-1)^2)^{3/2}} cdot frac{1}{f'(x)} = -frac{50}{(1-(5 x-1)^2)^{3/2}} The answer is -frac{50}{(1-(5 x-1)^2)^{3/2}}