suppose-a-is-n-times-n-and-the-equation-a-mathbfxmathbfb-has-a-solution-for-each-mathbfb-in-mathbbrn

Question

Suppose $A$ is $n 	imes n$ and the equation $A \mathbf{x}=\mathbf{b}$ has a solution for each $\mathbf{b}$ in $\mathbb{R}^{n} .$ Explain why $A$ must be invertible. [Hint: Is A row equivalent to $I_{n} ? ]$

Angelo Rendina · Accepted Answer

say we have the system A X equals B and disease Stoma solutions for every vector be in RN Now, if a he&#x27;s not in veritable, then we reduce the system metrics, eh? Me to something like upper triangle medics here and then on entire roll off zero at the end and appear this whatever. And then here we have the transformer coefficients so easy A is not in veritable. But that means that for some be hence be teal that the system cannot have solution against assumption that the system had solutions for every vector B. And of course, the problem is that we&#x27;re think we&#x27;re assuming that B is not in veritable so for the system to have solution always, the medics say, has to be in vertebral.

Runpeng Li · Answer

problem. Um, we want to show that if a X equals zero has only the trivial solution, she will solution. Then way. Want to show Dad? Hey, X, Because be must I have this ocean for every Be okay. Now, The way to do it without the convertible matrix serum is to observe the structure off A. So we consider let&#x27;s consider the reduced row ish in form off, eh? Say a prime. Okay, so scuse me. So that means, um, a prime times X equals zero. This system has has no pre variable because this system has has only has only intriguing solution that it means the system has no free bearable. Because we we know from our previous material we know that, uh, the system has nontrivial solution. It has at least a work repairable. So he&#x27;s just a negation of the statement. So has I&#x27;ll just write down here just for your information, A prime axe times. They&#x27;re a prime axe equal zero has has on Lee Trivial solution. It&#x27;s equivalent to say that this system has no free bearable. No, for you bearable. Okay, So for the Moore Demi&#x27;s matrix A has, um, has an people in positions. Of course. We&#x27;re assuming Matrix a had Matrix A&#x27;s On By and Square matrix. Yeah, this is our assumption. Um, people positions. Excuse me, people. Positions. And this is our conclusion here. Now, from here, we can say, um, since there are n people, people in positions and the matrix structure off, eh, Um, or we consider a prime and B and be seen as some some some values on the diagonal and zero for the lower triangular. An appetite for dinner. We have some numbers. I don&#x27;t. I just, uh, using darts to represent them. So, hey, in terms off a a prime time zags, we can see that of all the terrible here we are using a, uh, using this form of a prime then our nest our last variable se X end must be determined by the value we&#x27;re taking for B. So that means So we&#x27;re saying be one. So let&#x27;s suppose this is our vector B until fiend is equal to a prime times axe. And this is some sort of early on the diagonal times the vector X. So that means we were supposed defector ads his ex Juan x two txn Okay, So that means the last the last term Other day that I&#x27;ll be there diagonal If we multiply, multiply this This value by the nest. Last entry off. This factor acts. We have some. Aye, aye aye. Which is the entry for the last in the center of the diagonal of ape. Right? So? So there&#x27;s a II times the last the last available off the vector acts, which is X end is exactly the end. So we can solve for accent here. So that is bien divided by a II and so on so forth. So on so forth we can solve for everything Go bearable x one Until we were done, we have done for X one because, um, there&#x27;s no free bearable. So that&#x27;s why we can do this. So that means xn. That means acts the vector must exist. So what that means for the more damages the system a axe? It was B has a solution or every single b we&#x27;re taking the knob should be from are in

Angelo Rendina · Answer

we have the equation A minus. Hey, X Everything Immers equal Ex simmers me Now the first type is to show that bee is in veritable And to do so we multiply from the left by a minus I x so on the left hand side we&#x27;re left with identity and on the right Inside we have a minus a x ex embers Me But now the expression here in blue is by definition the inverse off me and so bees in vegetable. Okay, now that we know this, we can That&#x27;s right. Equation a minus x inverse equal X embers be by inverting both sides to obtain a minor sex equals being birth X. Remember that we invert the product. We get the product off the others in the inverse order. Okay, Now we can move the exxon aside. So let&#x27;s say we write a He&#x27;s equal to a plus being birth X. And now if we show that these metrics in brackets is in vegetable, that we can just multiply by theme verse to have axe equal to something and we&#x27;re done So to do that, we observe here that since a is in vertebral, we can multiply by eating birds from the right to obtain identity is equal to a Los beavers X a m birth. Now these means that X A chambers is the inverse Oh, a plus B in verse which is therefore is in vertebral. So now that we know that we can multiply from the left by the inverse off a plus by beavers And so we see a plus B in verse everything members, they&#x27;re multiplies, eh is equal to X and we have solved for X.

Angelo Rendina · Answer

So let&#x27;s go see a times B and seizing veritable So C minus one exists. So what we do here is we multiply both sides from the right by the inverse of C So we have C see in verse equal a B see embers. But now see, see in verse is by definition of inverse get entity metrics And on the inside we have a me CM birth Now about the associative property or matrix multiplication. We can put the brackets here and this formula is telling us that BC in verse he&#x27;s the inverse away so easy in vegetable.

Michael Jacobsen · Answer

in this example, we start out with this assumption that the Matrix A is of size and by end from there we&#x27;re going to start adding another condition. Let&#x27;s assume that the A matrix equation eight times X equals a zero Vector has Onley the trivial solution, So we have not said much about this matrix A. It&#x27;s a square matrix, and we know that X equals zero is the only solution to that matrix equation. From here we have some surprising consequences. Notice that when we say a X equals zero has only the trivial solution. That means there&#x27;s a unique solution, and this only happens if a X equals a zero Vector has no free variable. So now that we know that there is no free variable, we can say something further about this particular matrix. A. So this implies when there is no free variable that a has a pivot in every column. Let&#x27;s pause here for a moment with that statement and use the fact that a is a square matrix. So if a was of size three by three for Justin example to consider on the margin here and if we had a pivot in every column. Then there would be a pivot here, which we could force into a one here and here. But as soon as we find pivots in every column, it means that the pivot is along the main diagonal. And if there was any number here, say a four a row operation would eliminate it. Since we can place zero here and here, then there&#x27;ll be zeros above and below all the pivot positions. So this would be role equivalent to that to buy or three by three identity matrix. If a row equivalents to this matrix here. So this tells us then that&#x27;s a ISRO equivalent to the n by N identity matrix. But then, as soon as we know that it&#x27;s Roy equivalent to that matrix that immediately implies this matrix A is in vertebral that deserves an exclamation mark. Always started with was the assumption that the matrix is square of size in by in. And this equation has on lee the trivial solution. That&#x27;s all it takes to determine that the Matrix A is in vertebral

Pam Russell · Answer

so to be in vertebral, the inverse of a matrix Times itself has to equal the identity matrix. And so we want to prove that the identity matrix minus a is in vertical. So to do that, we&#x27;re gonna take the inverse of this matrix, multiply it by itself, and we&#x27;re hoping that will get the identity matrix. Now we were given that the inverse of this matrix waas the inverse plus a plus matrix, a squared plus matrix a cute we&#x27;re going to multiply that by our matrix. So, using our multiplication, we will take the identity times the identity that&#x27;s just going to give us an identity, the identity times matrix A. But notice that it is theon position of a so we&#x27;ll get negative a the identity times a And then we&#x27;ll have a times the opposite of itself. And that&#x27;s going to give us a square a square times. The identity will be a square, a squared times matrix a but its opposite that will give us a opposite. A cube, a cube times the identity will give us a cube in a cube times the opposite of a will give us A to the fourth Now we can go ahead and reduce some of this and simplify it down, and we get the identity minus a to the fourth. But we were given in the problem statement that aid to the fourth equaled zero. And so if we substitute that in, we get that is equal to the identity. And we have just proven that the identity matrix minus a is in veritable.

Problem

Exercises 25 and 26 prove Theorem 4 for $A=\left[…

Problem 24 Medium Difficulty

Suppose $A$ is $n \times n$ and the equation $A \mathbf{x}=\mathbf{b}$ has a solution for each $\mathbf{b}$ in $\mathbb{R}^{n} .$ Explain why $A$ must be invertible. [Hint: Is A row equivalent to $I_{n} ? ]$

Answer

So, that $A$ is a row equivalent to $I _ { n }$ . Hence it is invertible.

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Algebra Educators

Anna Marie V.

Maria G.

Kristen K.

Michael J.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 2

Video Transcript

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

Linear Algebra and Its Applications

Anna Marie V.

Maria G.

Kristen K.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

So, that $A$ is a row equivalent to $I _ { n }$ . Hence it is invertible.

Linear Algebra and Its Applications

Anna Marie V.

Maria G.

Kristen K.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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

Anna Marie V.

Maria G.

Kristen K.

Michael J.

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

Linear Algebra and Its Applications