Solving a Matrix Problem: Help Appreciated!

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Discussion Overview

The discussion revolves around solving a matrix problem involving polynomial functions of matrices, specifically finding p(A) for a given matrix A and polynomial p(x). The scope includes mathematical reasoning and clarification of notation used in matrix algebra.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a method for finding p(A) using the diagonal elements of the matrix, but acknowledges that this approach fails for higher degree polynomials.
  • Another participant challenges the idea of subtracting a scalar from a matrix, suggesting it is more appropriate to consider a scalar multiple of the identity matrix.
  • A different participant proposes a polynomial of the form 2x-2 and questions how this would apply to the matrix.
  • There is a discussion about the proper notation for matrices, with some participants emphasizing the use of capital letters for matrices and the importance of clarity in expressions like p(A) versus p(a).
  • Participants express uncertainty about the notation and its implications for solving the problem, particularly regarding the treatment of constants in polynomial expressions.

Areas of Agreement / Disagreement

Participants express differing views on the proper interpretation of the problem and the notation used. There is no consensus on the correct approach to solving the matrix polynomial problem, and multiple competing interpretations remain.

Contextual Notes

There is a lack of clarity regarding the notation used in the problem, particularly whether p(A) or p(a) should be used. Additionally, the treatment of constants in polynomial expressions when applied to matrices is not fully resolved.

Who May Find This Useful

This discussion may be useful for students studying matrix algebra, particularly those preparing for exams and seeking clarification on polynomial functions of matrices and notation conventions.

JasonRox
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I'm studying for a test and I got stuck here.

Let A be the matrix:

3..1
2..1

In each part find p(a).

a) p(x)=x-2

The answer is:

1..1
2..-1

The only way I can see this happen is that we take the numbers on the diagonal and pluck it in p(x).

So...
x-2=3-2=1

x-2=1-2=-1

Leaving the others alone, we get.

1..1
2..-1


The problem is, I know this is WRONG. This method does not work for b) and c), which are polynomials of higher degrees.

The inverse is:

1..-1
-2..3

I still can see a solution pattern.

I tried to relate to the chapter, but no luck.

Any help is appreciated.
 
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A matrix minus a scalar? That doesn't make any sense.
 
It is a matrix minus a scalar multiple of the identity matrix.
 
Let's say I choose 2x-2.

Would that be...

4..1
2..0

If this is correct, would x^2 imply multiplying both matrixes.
 
first of all do follow the standard notations ...
we usually use capitals for matrices ...
secondly
2X - 2
more often than not implies
2X-2I
where I is the identity matrix ...
so u can see why the answer is the way it is ...

and yes X^2 implies multiplication of matrices ... i.e X*X

-- AI
 
That's not what it says.

The question doesn't have capitals and refers to the matrix as A.
 
jason,
it also asks u to find p(a)
now this is really bad use of the notation actually.

p(x) =x-2 is fine
but i would expect them say
find p(A) and not p(a)

and yes when u sub in a matrix for x in an equation (in x) .. the constant (say c) is dealt as c*I ..

-- AI
 
I really appreciate your help.

I understand where you are coming from. This might be the reason why I didn't know what was going on.
 

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