5

(5) 2. Consider the following matrixFind det Find Cu the cofactor of entry 43"...

Question

(5) 2. Consider the following matrixFind det Find Cu the cofactor of entry 43"

(5) 2. Consider the following matrix Find det Find Cu the cofactor of entry 43"



Answers

Let $A$ be the given matrix. Use technology to calculate det $A$. $$ \left[\begin{array}{rrr} 17 & -4 & 3 \\ 11 & 5 & -15 \\ 7 & -9 & 23 \end{array}\right] $$

In discussion. We have toe find the determinant off the matrix. 50 minus three zero, 12 4 and 163 along the row or column indicated so. First we have to expand a long roll to So after expansion, we get something like the row multiplied by zero minus three six three minus 12 multiplied by determinant off five minus three one three plus phone while deployed by determinant off +50 one six. Yeah, between sweet. After expanding, we get minus 12, multiplied by 18 plus four multiplied by 30 which after evaluating we get minus 96 as the value off the determinant. Then we also have to expand along Column two, which means after expansion, we get something like this. Zero multiplied by determinant off 04 13 minus 12 multiplied were determinant off flight minus three one three plus six multiplied by determinant off five minus three. Cyril four which after expansion we get minus 12, multiplied by 18 again plus six multiplied by 20 uh which after expansion evaluating we get minus trying to success the value off the

In this question, we have been given the Matrix minus three one full 56 minus three and one and we have to find the determinant along the row or column that I've been given. So first year, according to question, we have toe expand a long row one So expanding along the straw we have minus three multiplied by determinant off 56 minus three and one. Then you have minus two determinant off 462 and one. Then plus one multiplied by determinant off four, five, two and minus three. Now expanding this we get minus three, multiplied by 23 minus two multiplied by minus eight. Then minus 22 which will give us the determinant off the metrics as minus 75. Now expanding along column too. That IHS, they have to expand the long this so expanding a long column too. We get minus three multiplied by determinant off five six minus three one minus four multiplied by determinant off to one minus three and one plus two multiplied by determinant Off to 15 and six. Now expanding this we get minus three, multiplied by five plus 18 minus for multiplied by two plus three plus two multiplied by 12 minus five, which gives us minus three, multiplied by 23 minus four, multiplied by five plus two multiplied by seven, which gives us minus 69 minus 20 plus 14. Yeah, it's uses the value off the determinant as minus 75.

In this question, we have toe find the determinant off the matrix minus two, four 71 than 3000 Then eight, five, then five, 10, 6. Seattle, five, fetal. So, after finding the determinant along by expanding a long road to I get three multiplied by determinant off 471 five. Then five 050 Then I ignore the rest. Because these are all elements, Cyril. So after expanding along brought three because I see that this will make my calculation simpler. I have three multiplied by five multiplied by determinant off four, one 515 So this will give me 15 multiplied by 20 minus five. It shall give me 15 multiplied by 15. So this will give me the value of the determinant as 225. What? Then we have to expand the long dark column four. So after expansion, I get minus one multiplied by 300 85 10, 605 Then the next would be zero. Then again, minus five multiplied by minus 247 300 Can I have six? Seattle five now expanding these along for the first matrix. I'll take grow one for the second metrics I'll take wrote to. So after expansion I get minus one multiplied by determinant off five, then 05 and the rest oft elements for this would be zero. Then I have minus five years minus five, multiplied by minus three, multiplied by determinant, off for seven, futile five. So to forgive me, there should be ministry. Actually, because I have taken the first row, the element is three so minus three multiplied by 25. Then I have plus 15 multiplied by 20 which would give me minus 75 plus 300 which would give me the well after determinant asked. 225.

This question is asking us to find the determinant of a four by four Matrix phasing co factor expansion. First on road to right here and next on column two right here. So let's start with Road to and I like to start left, right? So let's look at this floor. Well, first, we have to be careful about our signs when are doing co factor expansion. We know that the top left has a positive sign, and then you kind of alternate as you count out like that. And so we know that this four has that negative slam that we need to watch out for, so we'll do negative for and then we multiply it by the determinant of the minor that doesn't have that row and column. So not that row and not that column. It's all right. 006 night of 370 fine or two. Next, we'll look at this 13, and with that 13 we'll have a plus 13 and then the determinant of its minor. So six negative 35 negative 174 802 Then we'll have a minus six, and then we're just filling out the same the same kind of method skipping this row or that column in that row. Now we'll have 605 negative one zero form 862 and lastly, we'll have a plus negative eight. So I'll just write minus eight, because we were plus minus, plus minus. Plus, there we go. Just making sure and said The Associated Minor. Here we'll have a 60 negative three on top night of 107 and lastly, 860 Now you have to do this the second time with all these three way three matrices. I did a few calculations beforehand, but I'd recommend looking at this column here because it has a lot of zeros in it. Then for this one, perhaps maybe this bottom row Or maybe this column right here you can do an expansion on just I'd recommend doing an expansion on this column because it has a lot of zeros and and likewise the same one here. So when you figure out those three by three matrices and determinants, you will get negative four times negative. 282 US 13 times nine negative, 298 and then minus six times negative. 174 Nicely my s eight times negative. 234. Now if you multiply all this together, add them all up. Be careful about your signs. You should get 170. So this is how we do the expansion on road to now. Let's look how we do on column two. And so it'll be this one right here. Fortunately, this one has a lot of zeros in it, so it should be a little bit easier. We'll have 13 and this is positive because we can count help plus minus plus. So we have positive. 13 multiplied by the determinant of this minor, excluding its Roman column. So six negative 35 negative. One, 74 802 And then we'll also have a plus six and then likewise the values that aren't in the sixth row In column six. Negative 35 You have 46 98. Lastly, negative 174 And similarly, I did a few calculations beforehand, namely, the determinants of those three by three matrices and what you get here as you get 13 times negative negative. 298 us six times. 674 not also get to 170. And so this is our final answer. And we got it in two different ways.


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