Are these statements TRUE or FALSE and why?

[LaraCroft]

Hey,

Just stumbled into some linear algebra questions in my textbook...that I can't quite seem to work out...

Prove or disprove the following statements concerning 2 x 2 matrices:

1) If A^2 - 3A- 2I = 0 then (A-1) and (A-2I) are both invertible.

(so I got the determinant to be 0, which would mean that they are not, thus making it FALSE..is this right?)

2) If A = EB and E is elementary then B = FA for some elementary F.

...I wasn't too sure about this one...

Thank you...Neon Vomitt was the one who sparked my interest...so THANK YOU to you!

 

[slider142]

Each elementary matrix represents a linear operation that can be undone. Add some rigor for a direct proof.

 

[arnaldur]

Regarding the first question. To be able to factor the left side you must have +2I, so you have to add 4I to both sides to get (A-I)(A-2I)=4I.
If you then take the determinant you get det(A-I)det(A-2I)=4, so (A-I) and (A-2I) must be invertible.

 

[HallsofIvy]

Hey,

Just stumbled into some linear algebra questions in my textbook...that I can't quite seem to work out...

Prove or disprove the following statements concerning 2 x 2 matrices:

1) If A^2 - 3A- 2I = 0 then (A-1) and (A-2I) are both invertible.

Do you think the fact that A^2- 3A- 2I= (A- I)(A- 2I)= 0 is important?

(so I got the determinant to be 0, which would mean that they are not, thus making it FALSE..is this right?)

2) If A = EB and E is elementary then B = FA for some elementary F.

Every elementary matrix corresponds to some row operation and every row operation has a specific "inverse row operation".

The row operations are:
1) Swap two rows. And its inverse is itself: "Swap the same two rows"
2) Multiply a row by the number a. And its inverse is "Multiply the same row by 1/a".
3) Add a times a one row to a second row. And its inverse is "Subtract a times the same row from that row."

And, in fact, your statement isn't quite true because of a problem with (2) above. "multiply and entire row by 0" is a row operation although not used very often for obvious reasons! The elementary matrix corresponding to that row operation is the identity matrix with one row changed to all 0s. If E is such a matrix, there is no such F.

...I wasn't too sure about this one...

Thank you...Neon Vomitt was the one who sparked my interest...so THANK YOU to you!