5

2 3 L 4 1 1 8 1 1 3 1 aIe523809...

Question

2 3 L 4 1 1 8 1 1 3 1 aIe523809

2 3 L 4 1 1 8 1 1 3 1 aIe 5 2 3 8 0 9



Answers

$\left[ \begin{array}{rrr}{-3} & {1} & {0} \\ {0} & {-3} & {1} \\ {4} & {-8} & {2}\end{array}\right]$

In this question we have to use row reduction to find the inverse is of the given batteries if they exist. And check it by multiplication. No, let us consider the metrics. Yeah. 123, 4 01, 2, 3 0012 0001. And on the right side identity metrics. Or for the four 1000 0100 0010 0001. Now we will row reduce the all metrics. We will apply the operation are funny those two our than minus two. Our two stores too, uh minus Artie And our three stores too. Our 3 -R4. On applying these operations, we get the metrics 1111. Yeah 01 11 00 11 little little 01. And on the right hand side we get 1 -10 needle 01 minus one deedle needle needle one minus one 0001. Again. We will apply the operation are one stores too, Urban -R2. Our two stores too, Our 2- Artery. And our three stores too. Our three minus are full on applying these operations to get the metrics 1000 0100 0010 0001. And on the right hand side one minus two, one needle 0 1 -2, one digital hell one minus two 0001. So in investment taxes, one minus two, one hero 01 minus 21 001 -2 0001. Yeah. Now we will check it by multiplication. We will multiply A. And N. Was matrix. So we can write a Multiplied by a invest metrics equals two 1234 0123 beetle beetle 12 0001. Multiplied by in west metrics 1 -210 0 1 -2 one. You know the middle one by understood deedle deedle? They're all one. No. Yeah all multiplication. We will first multiplied by stroke with first column. So one multiplied by one plus two multiplied by zero Plus three multiplied by zero Plus four, multiplied by zero On simplifying it we get one similarly. Now we will multiply first row by second column one multiplied by -2 plus two, multiplied by one, three multiplied by zero Plus four, multiplied by zero and simplifying it. Be good feel Now we will write these values in the desire my tricks, €1. By following a similar method we will find the other elements of the metrics. So you know beetle 0100 0010 0001. Hence hey multiplied by and was metrics. It were to identity matrix. Thank you

This video, We're gonna go to the answer. A question of a 13 from Chapter nine White three for us to find the inverse off. The matrix minus two minus one 210 31 minus four. So let's combine this with the identity matrix once there is, There is there were once they were serious, they were What? Yeah. Reduce. So that's that three altitude of the first row to the bottom room. So that's going to go to zero. Ah, mine is 1/2 minus one ad for you, too, is 1/2. I want us to be over twos to you too. Keep a zero and one. And let's also add one of the first road to the second round. Get rid of this too. We're all scared of this wall. Uh, this becomes a wall 10 Top row stays the same minds to you. Minus one. Ah, whoa! 100 Next up, less subtract one of the middle row from Sapporo. So that's gonna be minus two minus one zero. Uh zero minus one zero. Minimo stays the same. 0011 What? Zero. That's also most black bottom are about to but zero minus 11 302 Next up, let's subtract one of the middle row from the bomber. That's gonna be zero minus one. That's zero at three months. Ones, too. They're minus one minus. Y T minus zero is too. Keep the middle. Where was it? Is seriously, Rabban. War hero. And keep the top roses minus two minus one. They were. They were minus one zero. Okay, let's subtract one of the bottom row from the top room. So I'll get rid of this month's one at zero minus two is minus two minus one minus 110 Uh, zero minus two is much too, kid made about the same there. Is there a woman? Rome on zero on dhe bomber. We can multiply by one, get zero. What? Zero. So most by my minus 1010 too, because minus two minus one equals one. T because minus two. Now, what we can do is most by the top row by minus, huh? It's gonna be born zero zero. Whoa, zero. What? Now you'll see that the form two rows. Uh, if we just flip these around, which you can do it, bro. production on. Then we get the I don't see a downside. So 010 That's what was the bottom room minus 21 minus two in the middle ground, which is not about tomorrow. 001 Well, zero on Dhe. This matrix here is the inverse off the matrix that we started with.

In this caution we need to calculate the universe of uh metrics A which is uh three by three matrix having entries in first row as 123 in second row, 0 to 3. And in third row 101. So first of all we will consider these metrics as having metrics A. As one too three and 0 to 3 Here. And here is the identity matrix of order three, 010 And 001. Now we will use the row reduction method to convert these metrics here to the identity metrics and then applying same operations to this identity metrics, we will get the metrics here as the universe of these metrics. So first of all we will um we will apply the operation to it as uh Artie stores are 3- Irwin. So we will get as our three stores. Our 3- Our win. Yeah we will get 123 here in the first round 0 to 3 in the second row. and here 1 0 will be Sorry 1 -1 will be zero And 0 -2 will be -2 and 1 -3 will also be -2. And this for this identity matrix this will be the same in faster and second role. Just 1000, 1.0 and zero minus one will be minus one here. Zero minus zero will be 01 minus zero will be one. So we got these medics. Now we will apply the operation on ah Row Arvin which is our ministers are 1 -22. So we will get as Applying this rule operation are one stores Are 1 -22 and we will we'll get here as one minus zero will be one to minus two will be 03 minus three will be zero again And writing the same the second row and taro has same. So 0 to 3 & 0 -2 and -2. And for this identity metrics being changed to one minus zero will be 10 minus one will be minus 10 minus zero will be zero And here 010 and -101. No. Ah now we will use the operation for the uh oh artie as RT plus are due Our three will store RT Plus R two and we will get So here applying the operation are three stores RT plus Or two. We will get here as For the first row 100. And for our to being same 0-3 in our three we will add a road to which will be zero plus 00 minus two plus two will be zero minus two plus three will be one. And for these metrics this will become 1 -1 0 1, 0 and -1 plus zero will be minus one and zero plus one will be +11 plus zero will be one. So we got these metrics. Now uh huh We will apply the operation to road to which we will apply road to minus Rodeo will store our two tries of our tree. So this will become and we will apply this operation road to stores roto minus tries of rotary and we will get here as 100 and here zero minus tries of zero will be zero to minus tries of zero will be 23 minus tries of one will be three minus three which is zero. Now writing the road hard seem so 001. For these metrics will get one minus 10. And here subtracting ties of our three we will get zero minus of minus three times minus one. So this will become zero plus three here as 31 times one minus tries of one will be one minus three. That is minus two. Do you know minus try? Someone will be -3 here. Ah And here eyes -1, 1 and one For the row three. No the only single element different from here from the identity matrix is to here. So we will divide road two by two. So we will this road to will store half of R. 02. So we will get here as 100. And here now zero by two will be 02 by two will be one and zero by two will be zero Against 001. And for these metrics we will get one minus 10 He had 3/2 and here minus 2/2 will be minus one and here minus 3/2 And rotary being same -1, 1 and one. Now we got these metrics as identity and this matters must be the universe of medics A. And hence we get metrics. Universe will be called to Medics 1 -10, three by two -1 and -3 x two -1. 1 and one. So this is the inverse of matrix A. Now we will check whether this universe is correct or not by multiplying medics here to its universe And hence the eight times a universe. This will be the metrics is one, 23 and 0-3, 101 times the universe of these metrics is 1 -10 three by two. My next one -3 x two Here -1, 1 and one. Now we will multiply these mattresses. So first of all we will multiply the first of a to the mhm mhm mm. First column of metrics universe and we will get here as one times one will be one and one times two times 3 x two will be three. So plus three and three times -1 will be -3. So this will become one and now we're deploying first row to the second column here. So this will become one times minus one, times minus one will be minus one Plus two times -1 will be -2. And here three times one will be plus three, So -3-plus 3 will become zero here. Now, multiplying this first row to the third column of these metrics, we will get uh one time zero will be zero and two times my street fighter will become minus three and three times one will be plus three. So now this will again be zero and multiplying this second road to the first column, second column, third column. Similarly as we multiply it in this first row for these elements of this first row, we will get it as 010 and again for the third row, multiplying by first column, second column and third column, we will get 001. So it is verified that the product of A and universes and identity metrics. So we got eight times a universe as identity metrics. Hence a university, the correct universe of medics. A. I hope all of you got discussion. Thank you.

So in this problem we're given this five x 5 matrix And were asked to use a matrix calculator, luckily in order to determine or find the determinant. All right, because you can do this by hand. I might be a little difficult. You can probably pick Grow four here and you know avoid two of the calculations. But then you're gonna still be left with four x 4 Matrix sees and working those ways where's goes down. So this can be a lot quicker. So we go over here to the new matrix. In our matrix calculator. We went to Desmond's dot com math tools matrix calculator to get this. And our matrix is a five by five. I gotta get this thing built up here, the framework of it five by five. and so the first entry is a three and then a minus two A four three. You know, one, Then a 1-0 -1 zero 2102 on Euro. And then 5 -15. My ass. One 032 zero three, two Than 4 7 -8. 4 seven. Mine it's eight and zero zero. And then we have 123 one two three 02 zero two chris Sanders. So that's all in there. Now I need the determinant of this Five by five. So I go d et determinant of A And there it is 410. That was a lot quicker and a lot easier. And trying to do this all out my hand, wasn't it


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