[Solved] 18.06 Unit 2 Exercise 10-differential equations and eAt

Problem 23.1: (6.3 #14.a Introduction to Linear Algebra: Strang) The matrix in this question is skew-symmetric (A^T = −A) :

^du = ^⎡ −0c 0c −ba _⎦^⎤ u or uu₂ = au₃ − cu₁₃

_⎣ ^dt b −a 0 u.

Find the derivative of ||u(t)||² using the definition:

||u(t)||2 = u12 + u22 + u32.

What does this tell you about the rate of change of the length of u? What does this tell you about the range of values of u(t)?

Problem 23.2: (6.3 #24.) Write A as SΛS⁻¹. Multiply Se^ΛtS⁻¹

to find the matrix exponential e^At. Check your work by evaluating e^At and the derivative of e^At when t = 0.