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3) If x=rcos0 and y=rsine , simplify the cxpression aftcrtaking the determinant...

Question

3) If x=rcos0 and y=rsine , simplify the cxpression aftcrtaking the determinant

3) If x=rcos0 and y=rsine , simplify the cxpression aftcr taking the determinant



Answers

Show that the determinant of any $3 \times 3$ matrix is equal to the determinant of its transpose.

Okay. We want to find determinative off a transport times. Be so what we have to do is and what's given to us is a determiner off A and B. So, um, so what we're gonna do is we have determined it off a transport transpose be. What you have to do is we know the transports a deterrent, A and B. So we're gonna make this to determine it of a transpose times determine it both be okay. And what's given to us is the terminal B. So we know what that IHS. We're gonna put a minus for here. And we know that determinant of a transposed is the same as a. Since they're giving us that, I can say this times minus four. So this is equal to three because they give us three. That's determine it up a times minus four. So the answer is minus 12. That's it, Cube.

Okay, Um, this problem is asking to find the determinant of a inverse off a basically. So this really is the easiest have seen out of all the problems so far this formula, you need to know this formula that the inverse of a determinant a is one over determinant. A. So all we have to do simply is no to determine A In this case, we know because that's what they're saying is three. So he's 1/3. That's really the answer. There's not not mawr to it. For some reason, they give us determine that B is equal to minus four. But they're not way. We don't need to do anything with that. That's it. So if you know the formula, you know, determined of a all you have to do is just plug the numbers 1/3. That's it. Thanks. Bye.

Okay. In order to find the answer for this, what I'm gonna do is since determinate off Whatever is inside square, I can always write it as a determinant of be in verse A be times determine it off Be in verse A Be okay, So trying to make it a smaller pieces and we know the determiner of a B And since we know what their formula is for determinant of being verse, I can make this Onley. I'm just gonna go focus on one of these because they're going to be equal. So this is I can write to determine it off, be in verse, times determinant of, and times determine it of be okay. And this whole thing has to be multiply by the same Think I'm not going to write that. So I know that determinant of B in verse. Since I have determined, let me write the determine of aid or giving us to termina A What is it? Let me look at the problem. It is three. So I know this is three times determined. B is minus four. Okay? And I know the inverse of determine is going to be one over minus four. Okay, so if I get this whole thing, let me just go back to this color. Black and I repeat this times because the whole thing is repeated again. Since these cancel, it's really three times three vehicle to nine. So the answer is nine. So what? I was trying to show you here that you got to break it down. What you know is determine of A and B is given to you, so you break it down, knowing the formula, you can easily get to answer. Okay. Thank you.

Okay, Um, we have a three by three matrix matrix A I wrote here that matrix a right. Here's a one a two, a three and transport of the matrix. We have to prove that the determinant of thes two are equal. So the way you do it is basically, you should know what the transport of the Matrix is, if not, go to the previous of videos. But basically, you get the row making my column, and vice versa on you could see this year. So I'm gonna find a determinant of this matrix. So I'm going to just repeat a one. The one see one here A to me too. See to. And if I go through finding the three by three determinant, I'm gonna basically right here is going to be a one be to C three plus a to B three C one plus a three b one see to minus a three. The two see one minus a one b three see to minus a to B one, C three. So that's a determinant for for a for transports were going to do the same thing here. We're going to write a one A to a three b one be to 33 and the determinant for this is going to be a one B two c three. You see already there's a pattern, right? Plus the one see to a three plus See one A to maybe three minus See one. Me too. 83 minus a one C two 33 and minus B one a to C three. So in order for this to to be equal, all the elements, all that has to be equal. So as you see here, a to B to C three is the same as a to B to C three, the B one. No different color B one C two a three is the same as B one. See to a three with these two. Are you cool? And then see one a two b three c one a two B three. Right here a three. B two c one one goes here Well, changing color again. A one B three c two. These two are equal and B one C three a two piece to vehicle. So what that tells you is that a We prove that a determined over A is equal to determine it off a transpose when the Matrix in this case isn't three by three. Okay, that's it. A little messy, but actually very good problem. Thank you.


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