In this video, we're gonna go through the answer to question number 19 from chapter 9.5. So we're asked to find the for a fundamental matrix of wth e differential equation X prime is equal to a X where a is given by this two by two matrix. So first gotta find the item values, they're gonna find their respective, and I come back to us and inform e fundamental waitress excites find aiken bodies. We need Thio, uh, find the values off which the determinant off a minus. Ah, I people's ear. And so we can ride it that as a servant off minus one. My, uh, one aids one minus, huh? On that seven of this is just as a top left element times by the bottom, right element minus bomb lifetimes Top. Right. So this is gonna be just explaining the buckets. B r squared minus one minus. AIDS just minus nine. So we're not equals zero, then. Ah, it's gonna be equal to plus or minus the square of nine, which is three. Okay, so there are two aiken values, puts a minus three. It's not gonna find it effective. So we've gotta look for a factor. You such that? A minus that I value times the identity matrix times by you. That's what you want for the first I can value is equal to zero. Okay, so what's that? Matrix world? A minus three. I gonna be minus four one aids minus two times by. You want deep zero. So this means that the first element of you one if that is just equal to one, then the second element is gonna be equal to four. And you can show that from either the top rope or bomber at both Make the same equation. Okay, for the next one, find a three i times by. Let's call it u two. I was there, So that's gonna bay two one day. Well, because by some back to you to zero that satisfy this you two, you could be equal. Thio. Well, if the first value one, then second gonna be minus two. Okay, then our matrix is gonna be for the first column. Is gonna be Theo first. I value time first better. Yes. I said eat of our first line by your times. Two times. By that, I could better So yeah, the 1st 1 This is gonna be e to the three tea. That's good. Before eating three team top right is gonna be well, tough, right? Actually, one time by eats the minus treaty. Okay. On bottom, Right. It's gonna be minus two e to the minus three t. That's how fundamental matrix.