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2 V 1 0 Ji 9 1 1 08 Jl 8 8 9 832 W 3 8 F 68 3 1 1 8 1 1 3 8 Hi 8 [ 1...

Question

2 V 1 0 Ji 9 1 1 08 Jl 8 8 9 832 W 3 8 F 68 3 1 1 8 1 1 3 8 Hi 8 [ 1

2 V 1 0 Ji 9 1 1 08 Jl 8 8 9 832 W 3 8 F 68 3 1 1 8 1 1 3 8 Hi 8 [ 1



Answers

$$\left|\begin{array}{cccc} 1 & -1 & 8 & 4 \\ 2 & 6 & 0 & -4 \\ 2 & 0 & 2 & 6 \\ 0 & 2 & 8 & 0 \end{array}\right|$$

So on this problem, we're given this matrix and were asked to use matrix capability of a graphing utility to find the determinant. So I went to Desmond's dot com and went to math tools matrix calculator and got this matrix calculator. So I have a four x four matrix. So I go new matrix And I got four rows and four columns And the first entry is a zero and then a minus three in an eight and that too. And then eight one a minus one and six And then I -4 in a six. Mhm and zero and a nine. And then uh minus seven, two, zeros zero, zero and 14. They had dinner now to find the determiner of this, I use the D E T button right here go determine it a matrix A And there it is 7000 441 7441.

They're. So for this exercise we have this vector B. And the subspace dovey generated by the one, V two and V three that are these vectors that are defined here. So basically we need to calculate the Earth a little projection of you on this space to view. And just remember remember this projection is calculated as the inner proud of the vector V. Each of the generators of this subspace dog. In this case the generators RV one, The two and 3. So we need to calculate the we need to calculate the inner part of me with each of the generator divided the score of the norm of the generators times degenerates. So these for the three vectors B two square plus the interpreter of B would be three. B three. Did the square of the norm of B. Three. Okay, so just to remind you a little bit of the geometric intuition of this, is that the view is generated by these three vectors. So what we're doing is projecting we on each of the generators and then some that together. So we want We t. v. one and V three acts as a basis. Actually in this case they are linearly independent so they form a basis for this. Yeah, subspace of you. So we're writing the in terms of this basis. So we're projecting projecting on this sub space. So let's calculate the correspondent values that we need. So in this case we would be one. The product of B would be to dinner product of the would be three. So this is equal two, one half, There is a constitute and this inner product is equal to zero and then the norms. So because this is the cost to zero means that we don't need this term anymore is going to be equal to zero. So we just need to calculate the score of the norms for B. two and B one. So for me, one square of the norm, remember that there is equal to the inner product of the vector with itself. And in this case this result in one and the inner approach of B two square is equal 2, 1 as well. So these are actually military vectors. And then we just need to put all together on the four. So behalf that the projection of the vector B on the subspace, our view, it's equals to 1/4 times 11 one plus the vector V two. That is equal to one, 1 -1 -1. After some. In these two vectors obtain the action solution that is one half times the vector, three, three minus one minus one. That corresponds to their thermal projection of beyond this subspace of you.

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

Okay, This problem, we have equals to falling. And it's also a three by tweet matrix. So we have negative. 310 and then zero negative. 31 and four Negative. Eight two. Now we can obtain the characteristic equation again. As follows. The same exact procedure as the previous problems. We have a negative Lambda cute minus four. Lambda squared minus five. Lambda minus two equals zero. So, Landau, one comma to come on three are gonna equal native one negative one. And negative, too. Now, for Lambda equals negative too. We want to solve the following equation for the Eiken victor. A plus two i times u equals zero. Then we'll obtain u equals T where Ty's every parameter, as always, times 111 And for Lambda equals negative one. We saw this equation a plus. I times you equal zero. And when we solve this, we will get you is equal to a T times 1/2 1 and two or equivalently team times one, two and four. Because this is just the Eiken victor skilled differently, scaled up. And so we have our answers


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