Question

Given that a square matrix A satisfies the equation A^2 -7A -I=0 Show that A is invertible and find its inverse.

Answer 1

is A a 2x2 matrix?

Answer 2

It doesn't say that

Answer 3

I typed the problem verbatim

Answer 4

\[A^2-7A-I=0\]

Answer 5

If it's not, I don't think this is enough to imply that it's invertible, since every matrix satisfies its own characteristic equation this is essentially the characteristic equation of a 2x2 matrix, which means it has two distinct eigenvalues. But I think if it's say a 3x3 this isn't really enough.

Answer 6

Maybe I'm over thinking this though and there's like some easier way to go about this haha

Answer 7

I don't know what eigenvalues are, and I have asked my professor about the eigenvalues before when someone on here helped me and started by trying to explain them. He said that until he explains it to us the solutions to our problems won't require comprehending eigenvalues.

Answer 8

Ok, I think I found something better then for you, we don't need to go so deep, sorry about that!

Answer 9

What we have covered are matrix algebra, matrix inverses, vectors, transpose of a matrix, lines and planes in this linear algebra course

Answer 10

Ok what we have to do is factor it and rearrange this equation, that will give you the inverse:

\[A^2-7A-I=0\]
here I add the identity to both sides and rewrite \(A=AI\) which is fine since a matrix times the identity is just itself.
\[A^2-7AI=I\]
Now I can factor out one of the A's
\[A(A-7I)=I\]
Oh hey, a matrix times another matrix equal to the identity matrix means that it must be its inverse!
\[AA^{-1}=I\]
So here you go:
\[A^{-1}=A-7I\]

Answer 11

Could it be that simple? I must be overthinking it too. I was thinking they wanted specific matrix components...... I was wondering how I was even supposed to decide what size matrix it was. Thanks

Answer 12

ONe thing....

Answer 13

Haha yeah, if there's any step in particular that you seem doubtful about I could help explain more, yeah sure anything.

Answer 14

How does this show that it is invertible?

Answer 15

Because at one point it is equal to the identity matrix?

Answer 16

Ok, so suppose I show you this matrix equation,

\[AB=I\]

Is B the inverse of A?

Answer 17

Or would I need to show that multiplication both ways results in the identity matrix?

Answer 18

AB=I, and BA=I is how you prove it is invertible right?

Answer 19

Yeah you could show that if you like, but once you know \[AB=I\] then you know \[BA=I\] since without even knowing anything we can prove it.

In this example if you want to show that, you can do that if you like as well, just do what I've done but factor out A on the right instead.

I'll give the proof that \[AB=I\] implies that \[BA=I\] in my next post.

Answer 20

I actually looked back at the work I did before I posted this question and I have \[(A-7)A=I\] does that seem like it is ok without the identity attached to the 7?

Answer 21

Not quite, because what does it mean to add a scalar to a matrix?

Answer 22

However it is common to just represent a scalar multiplied by an identity matrix as a so-called 'scalar matrix' so as long as you understand that the 7 you have written there represents a matrix with 7's down the diagonal you're fine.

Answer 23

I know this sounds silly but what does it mean to add a scalar to a matrix? You can't do that right?

Answer 24

Yeah you can't! That's why you have to have the identity matrix there, so really that 7 you've written is shorthand:
\[7 \implies 7I\]