suppose-a-is-the-3-times-3-zero-matrix-with-all-zero-entries-describe-the-solution-set-of-the-equati

Question

Suppose $A$ is the $3 	imes 3$ zero matrix (with all zero entries). Describe the solution set of the equation $A \mathbf{x}=\mathbf{0} .$

Michael Jacobsen · Accepted Answer

in this example we&#x27;re dealing with the three by 30 matrix A that&#x27;s provided here for such a matrix. We can ask interesting questions such as if we were to solve a matrix equation eight times X equals, say, zero vector. What would be the solution set to such a system? Well, this is kind of a strange deal, since our Matrix A has nothing but zero entries. So if we go to the augmented matrix for this system, then it would have this form. First. I&#x27;m inserting a as green. So there&#x27;s a three by 30 matrix and I&#x27;ll use blue for the augmented portion. So there it is. This is the augmented matrix for eight times X equals zero. Interestingly, the variables are x one x two x three, and we have that x one x two x three are all free variables, since if a is zero matrix, there is just no pivot columns, and that makes all variables free in this situation. Well, in this case, if we were to describe the solution sets to eight times X equals zero matrix, the solution would be X, which is equal to x one x two x three and if we write this in Parametric Victor form, it takes on the form of X one times 100 plus x two times 010 plus x three The last three free variable time +001 where x one x two and x three are in our the set of real numbers. So this is the solution set in Parametric Vector form. But possibly this is just a descriptive f x one x two x three could be any real number whatsoever. And this is the solution, then the solution set to our matrix equation is Theo. Entire set are three. So that&#x27;s the solution. Sets to a X equals zero when we&#x27;re dealing with the zero matrix.

Jingyun Wang · Answer

the question asked to find a solution off the given system First, when you to change a system off equation into augmented metrics form. Remember, we need to find a reduced role are shown on form to find the answer for Joe to, we can do to times a role one plus a role chew for those three we can do grocery minus rule. Waas. Have your old one to the elementary rule operation for Row two and a rosary switch your rosary was wrote to for Grocery. We can do nine times rolled true minus roastery hobby. Other rules every time where we see all there was in wire. Oh, which means there&#x27;s a free variable for days. Case accessory is the food terrible for No one we can do Row one minus three times, wrote Chew. Now we have the reduce a rational form so we can get X Y equals zero x two equals zero X ray can be any number, and this is your final answer

Jingyun Wang · Answer

the question. I have to find a solution off the given system. First, we need to change your system off the equation into automatic of metrics for remember, When did you find a reduced her or I shall conform to find the answer. For euro two, we can do three times the rule one minus a road to grocery. We can do to times Tarawa class Roastery Copyright one to the elementary operations for Road two and a rosary every time when we see all there was in one row, which means stairs of favorable for this case acts two accessory and x four other food verbals and this is the reduce a rational form so we can get X one equal three times x ray. Thanks to you can be any number X ray can be any number X four can be any number and this is the final answer

Ankit Gupta · Answer

in this question. I have been given three exit caldo. It bless. Could he be where a and B mattresses are given to me as in the push in So clearly I will say from here three x is equal to a plus three b If I have to write blue minus two minus one 10 three minus For this is the A metrics plus three b metrics will be up. Tenders 09 60 minus 12. My industry on adding this are bigger ting mine Estelle it seven zero minus nine on this is minus seven. So this is the value off three x So if I have to solve for X, I&#x27;ll be dividing both sides by three. So that will get me minus two by three eight by three seven by three Muzito my industry and minor seven by tree. So that is the value the matrix X I over. It&#x27;s clear. Thank you

Jingyun Wang · Answer

the question I asked You find a solution off the given system. First, we need to change a system off equation into a semantic metrics. For remember, we need to find the reducer or rational for me to find the answer. So for row One, we can do road to minus 01 copy Row two Furrow to we can do three times Roll one minus wrote you kef usual one furrow to we can do 1/11 times wrote you for no one we can do roll one minus five times wrote you This is the reduced arsenal form so we can get X one equals zero acts two equals zero and this is your final answer.

Michael Jacobsen · Answer

in this example, we&#x27;re considering the homogeneous equation. X equals zero, and we&#x27;re told that we have a nontrivial solution right out of the gate. It&#x27;s X equals 111 Let&#x27;s suppose also that is, of size three by three. And the question is four. What matrix A is X equals 111 A solution to our homogeneous equation. Well, in order to determine such a matrix, say, Let&#x27;s first right out a X equals zero in Amore expanded form. Let&#x27;s put in a matrix. Say, I&#x27;m going to leave it blank For the moment, X is going to be 111 and this has to be said equal to the zero Vector, which is 000 Now we could make a completely the zero matrix, but that feels as though we&#x27;re cheating. Really, we wonder, Can we do this and get 111 a solution when a is not the zero Matrix? What one thing we can do is just because it doesn&#x27;t have to be. The Zero Matrix doesn&#x27;t prevent us from zeroing out a row or two, but I wouldn&#x27;t get too out of hand. I want to make the matrix a as simple as possible. So I&#x27;m going to make this 111 in this column. Now, if we think about what would happen if we multiplied out this much, it would look like this so far, we would take the vector entry here, which is a one, and multiply it with the first column of a So will be a column, 111 going here. We still have to decide what should go here. But we know that whatever we put will be plus one times those entries where I left question marks and for the last vector will be putting plus zero time. Excuse me. Plus one times that zeroed out column and all of this has to be equal to zero. So this is the format so far. And if we decide what can go here and then here, ultimately the vector equation to produce all the zeros will be all set will notice. That assumes we chose ah one here. If we put a negative one here, then add up, We&#x27;ll you will definitely get to zero, especially since there&#x27;s no contribution here. Some going to put negative one here and here and here in the corresponding vector equation. We then have a solution, and that tells us that this matrix A allows 111 to be a solution so effectively. If we make one column, the combination of another and zero out a column, that would be one way to determine such a matrix a.

Problem

If $\mathbf{b} \neq \mathbf{0},$ can the solution…

Problem 27 Medium Difficulty

Suppose $A$ is the $3 \times 3$ zero matrix (with all zero entries). Describe the solution set of the equation $A \mathbf{x}=\mathbf{0} .$

Answer

The solution set is all vectors in $R^{3}$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Catherine R.

Anna Marie V.

Heather Z.

Absolute Value - Example 1

Absolute Value - Example 2

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The solution set is all vectors in $R^{3}$

Linear Algebra and Its Applications

Grace H.

Catherine R.

Anna Marie V.

Heather Z.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Catherine R.

Anna Marie V.

Heather Z.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications