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Let Abe square matrix. If det(A) = 0 then the equation Ax = b has either no solutions or infinitely many solutionsTrucFalsc...

Question

Let Abe square matrix. If det(A) = 0 then the equation Ax = b has either no solutions or infinitely many solutionsTrucFalsc

Let Abe square matrix. If det(A) = 0 then the equation Ax = b has either no solutions or infinitely many solutions Truc Falsc



Answers

Deal with the problem of solving $\mathbf{A x}=\mathbf{b}$ when $\operatorname{det} \mathbf{A}=0$ Suppose that, for a given matrix $\mathbf{A}$, there is a nonzero vector $\mathbf{x}$ such that $\mathbf{A x}=\mathbf{0 . ~ S h o w ~}$ that there is also a nonzero vector $\mathbf{y}$ such that $\mathbf{A}^{*} \mathbf{y}=\mathbf{0} .$

Alright for this problem, we are given a linear system. A ax equals B where the reduced row echelon form of the augmented matrix a augment B equals now going down each row than across each column. 100 All right, 0 to 000 0100 3200 1400 Augment Negative 2500 So one thing that we can note is that the first column the third column have leading ones there and there, whereas the 2nd, 3rd and 4th do not. So what that tells us is that we're going to end up having three parameters in our solution. So translating that first expression into a equation give us that X one plus to actually what I'll do is, since we know that we have three parameters, we'll say that off the bat X two equals R X four equals S and X five equals t, then translating the two non zero rose that we have there we have that X one plus two are plus three s plus t equals negative too. And we have that X three plus two s plus fourty equals five now what we can dio So we can rearrange these expressions and turn them into matrices. So the meat or not, just matrices rather vectors instead, or a sum of vectors. So what we would do is we would want to go through. And essentially we are first rearranging and figuring out what x one and x three are in terms of S and T as well as their Constance. So we would have that X one equals negative too. Minus two are minus three s minus. T X three equals five minus two s minus 40. So that gives us the vector formed. First of all, we'll have our vector holding all of our constants. So we have that the constant part of X one is negative two. Then there's no constant part of X two. Should be a four by one or sorry, five by one. Rather, um so x two has no constant part. X three has constant. Part five x four has no constant part, and x five has no constant part. So is the constant part of our vector. Then we'll have our our component so we can go through and figure out the our parts so x one has negative to our we said that X two equals our lips. I should just write negative to their X four are started. Rather X two equals are so we have a one in that place and then zero for the other. Actually, you have in X three. Yes, we have zero ours in x three. None in x four x five. So we get negative. 21000 Then we would have plus s. Now we go through and find our s terms. We have three in, um, rather negative three in X one. We'll have zero and x two is x two is just are you'll have negative two in x three. That guy, we have positive one in X four from the definition and we'll have zero in x five. And lastly, we have our tea component, which will have negative one from x one zero from X to negative four from X three zero from x four and one from X five. So that gives us our solution or all possible solutions. Yeah, to that system of equations where we have further specify that r s n t are real numbers next we want to consider if the first column was replaced by 1134 and the third column was replaced by two negative 113 And we were still doing a X times be just with this modified matrix. Then what is B s? So we have, you know, a one a two a three a four a five where a one and a three have been changed. So right, a little prime on them, then A to a four and a five are the same and then we're multiplying them by the same X factor. What's that going to be? So first, what we'll do is write out what that matrix looks like. I'll pause so our modified matrix looks like 1134 2000 to negative. 113 32001400 And then we're multiplying by the X that we found in the previous part. So it's multiplied by negative two minus two ar minus three s minus T are five minus two s minus 40 s t turning our some of vectors the way that we expressed s or ex rather in part a turning that some into a single vector here. Then we have to do our matrix multiplication. So keep in mind that the way that the matrix multiplication works is that we go across, we take the cross product of each row vector with the column vector. And in this case, since we're doing a multiplication of a matrix by a vector, we go through and each component of the resulting vector is going to be the cross product of one row with the vector. So we would have the first or sorry. I said cross product, I meant dot product rather s. So we have a one dot Exe the first element a two x the second a three got x who, actually important thing to be careful about here. In terms of the notation, I believe that with the notation the textbook uses, we would right a one a two a three a four like this to refer to the row vectors. Whereas the arrows overtop are being used to refer to column vectors. I'm just going to confirm that yes, So there is a convention that the arrow overtop is used to designate that something's a vector, but in this case, when we're saying a one a two, a three and a four were saying the, um we're saying the row vectors, whereas using the arrows when referring to Indices of the Matrix, is referring to the correspondent column vectors. So that being said, we have a one dot exe a three x a four x So a one X is three dot product of 12231 one with negative to negative two ar minus three s minus t are five minus two minus 40 an s than tea. Which do you excuse My sloppy writing here going to equal s. So we have one times the first row. So we'll have It's not going to be a vector anymore. It's gonna be a scaler. We'll have one times The first row is going to be just negative. Two minus two are minus three s minus T. We have two times a second row which is just going to give us plus two are Then we have two times the third row, which is going to give us 10 minus four s minus 80 and we have three times the fourth row which three times the fourth row, which is going to give us plus three s. And then we have plus T for a to not ex and so on. What I'll do is, you know, I've gone through this in specific breaking it down and for the other elements I won't be as I won't be showing it as step by step like that. We can copy down that first element here this copy and paste that up there for the second. Um, for the second row of the resulting vector, we should get again. Negative. Two minus two ar minus three s minus T. And no, ours from the zero times are and we'll have negative one times five minus two s minus 14. We'll have negative five plus two s plus 40. We'll have two times s or plus two times s. Rather, we'll have plus four times t. Yeah. Then we'll have three times negative. Two minus two ar minus three s minus t. So we'll get negative. Six minus six R minus nine s minus three t. Then plus zero times are then plus one times five minus two s minus 40. Then we'll have for the next row we'll have four times Negative. Two minus two ar minus three s minus t. So we'll give us us will give us negative eight minus eight are minus 12 s minus 40 than plus three times five minus two s minus 14. So it will be 15 minus six s minus 12 t which when we simplify that down, we should get moment here. Lee, pull it up on neither screen. We should get that. The most simplified form of what we got there. It's negative. Two plus two times five minus two s minus 40 then negative seven. It's too are actually. One moment I realized, as I was starting to copy it down that I could have simplified it further. So the most simplified form actually be eight minus running alone. Battery eight minus four s minus 80 than negative seven minus two are plus s plus 70. Negative one minus six are minus 11 s minus 70 and seven minus eight are minus 18 s minus 16 T. Where again, R S and T can be any real number

We have given the medics here 12 and minus four zero A. It was to be so Now we have to find the Here they come in all the willies off a B for which all went in my take says give a number of solutions So first exactly one solution. So that means only one solution One solution. So that means this is X plus two y equals two minus four and this is minus a white equals minus B. This is only be so see like wants to be the very with minus with the virus a devalue x plus This will Yoda two multiplied with minus B Divide with a equals two minus four will give X equals two his four plus to be divided with a that X equals two so that with 40 blessed be the red with a so any out of all I can say that hit any of the form here X equals two minus be divided with a and why equals toe food A plus to be the head with a will give exactly one solution. And now for the second part infinitely many solutions. So that means both are same so that you can see it. X plus two y minus a way he questo here. Zero. So this is ending this with very little it questions. It'll will you. We'll give infinitely Minister listen, So that means here, X plus y that is two plus a quick little So any question on the form next equals toe to plus a why will give infinitely many solutions. And if it gives a false statement that is here a here it and we are not equal. So that will give not any solution. That is, if a North because Toby, that will give inconsistent system off the listen.

This is a little question Number 63 as we can see here when a 0.0, then the last ruler of zero. And there would be a no pilot entry at the third column, right? When is equal to zero. And the system has infinitely many solutions then financially Manisha, right? No. When is not equal to zero then against that? That the system is consistent and stuff has only one solution right on the one solution that is negative. 25 and zero. Right. Thank you.

In the question it has given that E X is equal toe be where X is the unknown and he iss in vertebral. So this would mean that go find X. I can write this equation as a inverse multiplied by be and so


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