# Please help, do not know what this type of problem is called or how to solve it

• steph_swords
In summary, the nxn matrix B defined by the equation B = A2 - 2A +3I is invertible and find its inverse.f

#### steph_swords

Suppose that A is an nxn matrix which satisfies the equation

A3 - 2A2 + 3A - I = 0

Show that the nxn matrix B defined by the equation B = A2 - 2A +3I is invertible and find its inverse.

Does anyone have any idea what this type of problem is called, and what steps you take to solve it?

It's a matrix algebra (linear algebra) problem. Try factoring it like you would a normal algebra problem. You'll also need to recall the definition of a matrix inverse to solve the problem.

can you be more specific? for some reason i just can't see how to do this, I've literally spent over 2 hours attempting to solve it.

You need to tell us what you've tried and where you are stuck.

i think what's messing me up is the fact that there are equations involved. i understand how to do the inverse of a matrix, but I can't seem to connect the dots between the idea of a matrix and how it relates to the given equations.

i think what's messing me up is the fact that there are equations involved. i understand how to do the inverse of a matrix, but I can't seem to connect the dots between the idea of a matrix and how it relates to the given equations.

Try to recognize how the two equations are related to each other.

OK, so the conceptual block is the fact that matrices can be combined with operators to make expressions. The rules are almost the same as with normal algebra. The following is a matrix algebra equation that says a matrix B added to a matrix A equals matrix C:

A + B = C

Matrices can also be multiplied by scalars (meaning each element of the matrix is multiplied by the scalar):

A + 2B = C

Here are examples of an expressions using matrix multiplication (and addition):

AB = C
AB + C = D
A2 = AA
AB = I

In the last case, what can you say about B?

Also, matrix mutliplication is distributive:

A(B + C) = AB + AC

Is none of the above familiar?

I recognize all that. In the last case, B would be equal to the inverse of A. I know there is just something so small that I am missing, and I know I'll feel like an idiot when I see it. do i have to solve for A in the first equation so that I have an A= and a B=? idk I'm just frustrated, this stuff usually comes so easily to me, for the first time I understand what it's like to not be good at math.

Suggestion: solve the first equation for I and factor the opposite side.

Let's just abuse some notation for a minute and say $I$ is $1$. Carrying this out would give you

\begin{align*} & A^3 - 2A^2 +3A - 1 = 0 \\ & A^2 - 2A + 3 = B \end{align*}

Do you think you can make a simple substitution here?

you can factor out an A from the first equation and be left with A(B) = 1, am I on the right track?

Yes. And what does that tell you about B?

that it's equal to the inverse of A.

Yep. Now you should be in a position to answer the original questions.

okay I want to say I get it but I think my issue here is I don't know what format answer I am looking for. Is the inverse just going to be in the form of an equation as well?

okay I want to say I get it but I think my issue here is I don't know what format answer I am looking for. Is the inverse just going to be in the form of an equation as well?

What is wrong with saying that the inverse of B is A? After all, that is what you have just shown, and it does answer the original question completely.

RGV

yeah that would work, I guess it's one of those times the simplest answer is the right answer. thanks for all your help.

From $A^3 - 2A^2 + 3A - I = 0$ we can get $A^3- 2A^2+ 3A= I$ and then $A(A^2- 2A+ 3)= I$. Do you see the point?

you can factor out an A from the first equation and be left with A(B) = 1, am I on the right track?

Yes, you're on the right track, although that should be AB = I.

Now postmultiply both sides by $B^{-1}$.