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# matrices watch

1. Let A be an n × n matrix which has the property that A^2 = A.
(i) Write down the most general polynomial in A.

what does it mean by polynomial in A?
2. (Original post by swagadon)
Let A be an n × n matrix which has the property that A^2 = A.
(i) Write down the most general polynomial in A.

what does it mean by polynomial in A?
What does a polynomial normally look like?

What happens if you try to write down a quadratic or a cubic or something of higher degree? So what's the most general polynomial you can write down?
3. (Original post by davros)
What does a polynomial normally look like?

What happens if you try to write down a quadratic or a cubic or something of higher degree? So what's the most general polynomial you can write down?
would it be just A=0? i dont really understand what it means by most general
4. (Original post by swagadon)
would it be just A=0? i dont really understand what it means by most general
No, a general polynomial in a variable x would be something like or

What would happen if you tried to write down a polynomial like this in the variable A, given the property of A you've already proved? Is there such a polynomial as , or could this be simplified further? So what sort of polynomials are you left with?
5. (Original post by davros)
No, a general polynomial in a variable x would be something like or

What would happen if you tried to write down a polynomial like this in the variable A, given the property of A you've already proved? Is there such a polynomial as , or could this be simplified further? So what sort of polynomials are you left with?
would it be just aA+k? where a is a constant and k is arbitrary matrix, because all higher powers of A reduce down to A?
6. (Original post by swagadon)
would it be just aA+k? where a is a constant and k is arbitrary matrix, because all higher powers of A reduce down to A?
close.

aA+kI (where a,k are constants & I is the identity matrix)

This is from:
then why? think!
7. (Original post by swagadon)
would it be just aA+k? where a is a constant and k is arbitrary matrix, because all higher powers of A reduce down to A?
tombayes has given you the correct format - I was being a bit sloppy with my notation because my 'c' should have been 'cI' where I is the identity matrix

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