Are these statements TRUE or FALSE and why?

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Discussion Overview

The discussion revolves around the validity of two statements concerning 2 x 2 matrices in linear algebra. Participants are tasked with proving or disproving these statements, which involve concepts of matrix invertibility and elementary matrices.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the determinant being zero implies that the matrices (A-1) and (A-2I) are not invertible, thus claiming the first statement is FALSE.
  • Another participant argues that to factor the expression A^2 - 3A - 2I, one must add 4I to both sides, leading to the conclusion that (A-I)(A-2I) = 4I, which indicates that both matrices are invertible.
  • A different participant notes that the second statement regarding elementary matrices is not entirely true, citing a specific case where an elementary matrix corresponding to multiplying a row by zero would not allow for an inverse elementary matrix.
  • This participant elaborates on the types of row operations associated with elementary matrices and their inverses, emphasizing the importance of understanding these operations in the context of the second statement.

Areas of Agreement / Disagreement

There is disagreement regarding the first statement's validity, with some participants asserting it is false based on determinant calculations, while others argue it is true based on factorization. The second statement also appears to be contested, as one participant challenges its truth based on specific cases of elementary matrices.

Contextual Notes

Participants express uncertainty about the implications of the determinant calculations and the conditions under which elementary matrices operate, particularly in relation to the second statement.

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

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