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Givenanjcompule A Ae aJ lo wer (i0vjuler mAfri X G j #ppd Ia~J Ll( ( m4^ U Such I6t 8 .LV IMG 5659 jpg Sgarch4 3A-| -B-[4Com pufe A-i Ab and a (QWe traingua matrix ...

Question

Givenanjcompule A Ae aJ lo wer (i0vjuler mAfri X G j #ppd Ia~J Ll( ( m4^ U Such I6t 8 .LV IMG 5659 jpg Sgarch4 3A-| -B-[4Com pufe A-i Ab and a (QWe traingua matrix L cna Z1n Upper traingular matn * Such tnat 65 Lu

Given anj compule A Ae aJ lo wer (i0vjuler mAfri X G j #ppd Ia~J Ll( ( m4^ U Such I6t 8 .LV IMG 5659 jpg Sgarch 4 3 A-| - B-[4 Com pufe A-i Ab and a (QWe traingua matrix L cna Z1n Upper traingular matn * Such tnat 65 Lu



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Suppose that $$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \text { and } \operatorname{det}(A)=4$$. Compute the determinant of each matrix below. $$\left[\begin{array}{rrr} g & h & i \\ -4 a & -4 b & -4 c \\ 2 d & 2 e & 2 f \end{array}\right]$$

Different today. We're going to solve problem number 33 here to do Tom and is given by one. Okay. Zero lot determinant off is given the uh huh. The see the so he will be equals a plus. Casey B plus Katie seat The determinant off e equals a press case into the minus B plus Katie into city 80 plus k c V minus BC Linus K. D. C. So it it's like a d minus b c. Then this is a This is it determinant off equals what return mint off equals a T minus busy So determinant off e equals determinant off into the dominant off.

Great. We're going to solve a problem. Number the foot here. The government is a because mhm b c think it takes equals the 011 Seattle So determinant mhm equals zero minus one. The physicals minus one. So then e equals 0110 into a B C B, which will be getting it does seat the mhm the so determinant off. Yeah, equals C B minus 80. What we should be C minus 80 determinant off. Hey, because a T minus b c so determinant off and in the pit dominant off mhm equals minus learn Hindu A T minus B C to be B C minus 80.

Okay, so we're asked to use the fact that the determinant of matrix A is four to figure out what the determinant of the new matrix would be. Um, so I'm going to start with four. So I'm gonna write, determine it. New is four. And then we're going to discuss the changes that I see. Um, first of all, to get row one, all we have to do is take negative one times wrote to and added to row one. Well, that has no effect on the determinant according to postulate one. Okay, so, uh, let's look at Roe Three, the new road three in the new row three. That is negative. One times the old road to so we've multiplied a row by a scaler. So according to postulate to, we have to multiply by negative one right now. That would change the determinant by model plane by negative one. Okay, we look at the new row two and we see that that's the old row three just multiplied by two. Well, when you multiply by a scale or again according to postulate to, it's just ah get the multiplied by that too. In the determinant, we also switched Rose two and three. According to postulate one If you switch to Rose, then you have to add Put a negative in there. It's the opposite. Okay, Final thing we did is we just multiplied the whole matrix times three. So looking for which postulate this is possible it four The determinant of K A is K to the end power time The determinant of a So it is going to be K which is three to the end power. This is a three by three matrix. So three Okay, so negative times negative is positive four times two is eight So that's going to be eight times three times three is nine times three is 27 8 times 27 and eight times 27 is 216. The new determinant is 216.

All right, so we're starting from Matrix A, and we're going to use elementary row operations to change it into the second Matrix. So the first thing that I noticed is that row one if it is multiplied by negative four. Well, then it looks like that row. And so we're gonna multiply by Negative for postulate to says, If you multiply any row by a skill, er okay, then the determining of the new Matrix is K times the old determinant. So that's going to require us to write down the old determinant which is four. And then we need to multiply by negative. For I noticed that this row and this row are exactly the same. It's just multiplied by two. So I should put in a lot of play by two. I wish I would have written this in green. So there it is, its head in their race, that Okay, now, if you switch Rose according to postulate one, then the new determine it is the opposite of the old determinant. So I see that row three would have to be switched with row one in order to get it up there, so that would be one switch. So I'm gonna put a negative down here. And now I also notice that the other two rows to road to would have to switch with row three with the new row three, so it would be another switch. So two switches, um, switch one in three, then switch two and three. And that would put everything in the correct place. Just verifying that my reasoning was correct. Yeah. So our answer is the original. Determine it. Times negative. Four times. Two times negative times. Negative. So negative times. Negative times negative is negative. Four times four is 16 times two. It's 32 negative, 32.


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