1. ## Matrices problem-

Consider the matrix M = 3 -1
4 -1

Verify that M^2= 2M -I

the question is what is I ??

2. Originally Posted by Khonics89
Consider the matrix M = 3 -1
4 -1

Verify that M^2= 2M -I

the question is what is I ??
Generally, when talking about matrices, I is the identity element. ie. the element such that for any matrix A, A*I = A.
It's the integer equivalent of 1 under multiplication.
ie. if A is an integer A*1 = A.

For 2x2 matrices the identity has
[ 1 0 ]
[ 0 1 ]

for 3x3s it's

[1 0 0 ]
[0 1 0 ]
[0 0 1 ]

And so forth. I don't know how to draw matrices in this thing... sorry...

3. Originally Posted by Khonics89
Consider the matrix M = 3 -1
4 -1

Verify that M^2= 2M -I

the question is what is I ??
I assume that you can verify the given result. Then:

$M^2 = 2M - I \Rightarrow M^{-1} M^2 = 2 M^{-1} M - M^{-1}$ $\Rightarrow M = 2I - M^{-1} \Rightarrow M^{-1} = 2I - M = \, ....$.

Side note: $M^2 = 2M - I \Rightarrow M^2 - 2M + I = 0$ has been got using the Cayley-Theorem Theorem: Cayley-Hamilton Theorem

Edit: Whoops, I just realised that your question wasn't find the inverse of M. Not to worry, since you asked what I is I figure I've probably anticipated what your next question is.

4. Surely that's more complicated than necessary.

If $M^2= 2M- I$ then $M^2- 2M= M(M- 2I)= I$ so $M^{-1}= M-2I$ by the definition of "inverse matrix"- its product with M is the identity matrix.

5. First square M and you get:

5 -2
8 -3

Then multiply M by 2 you get:

6 -2
8 -2

Finally take away the identity matrix I away from 2M:

|6 -2| - |1 0|
|8 -2| |0 1|

Whcih gives you the same valve as M^2..........

6. Originally Posted by HallsofIvy
Surely that's more complicated than necessary.

If $M^2= 2M- I$ then $M^2- 2M= M(M- 2I)= I$ Mr F says: *Ahem* .... = -I, surely ....
so $M^{-1}= M-2I$ by the definition of "inverse matrix"- its product with M is the identity matrix.
Well, one point in its favor is that the more complicated way appears less prone to error .....

7. Don't you have a "smilie" for "abject surrender"? It would fit here.

(I considered "mooning" for a second but thought better of it.)