# Is A^2 equivalent to AxA or all the elements of A are squared?

1. Feb 8, 2010

1. The problem statement, all variables and given/known data
I was just wondering if say A is a 2x2 matrix. Is A^2 equivalent to AxA or all the elements of A are squared?

2. Relevant equations
let A and B be 2x2 matrices. is the following true?
(A-B)(A+B) = A^2 - B^2

3. The attempt at a solution

2. Feb 8, 2010

Re: matrices

For 1: Raising a square matrix to a positive integer power is defined exactly as it is for numbers: $$a^2$$ is short-hand for the product of $$a$$ with itself.

For 2: The rule $$(A-B)(A+B) = A^2 - B^2$$ is true for numbers because multiplication of reals is commutative. FOIL out the left side, and look for the spot where that property is used for real numbers in order to get to $$A^2 - B^2$$. Is the corresponding statement true for matrix multiplication?

3. Feb 8, 2010

Re: matrices

does that mean A^2 = AxA?
erm.. yes?

4. Feb 8, 2010

Re: matrices

Yes, A2=AxA

5. Feb 8, 2010

### HallsofIvy

Staff Emeritus
Re: matrices

Yes, A^2= A*A.

No, (A- B)(A+ B) is not equal to A^2- B^2. It is equal to A^2+ AB- BA- B^2 but the AB and BA do not cancel because matrix multiplication is not commutative.

6. Feb 8, 2010

Re: matrices

ahh I see :)
I cancelled out AB-BA ignoring the fact that matrix multiplication is not commutative
thanks!

then I suppose (A-I)(A^2 + A + I) = A^3 - I is true?

attempt : expand LHS

= A^3 + A^2 + A - IA^2 - IA - I^2
= A^3 + A^2 + A - A^2 - A - I
= A^3 - I (since I^2 would be I right?)

Last edited: Feb 8, 2010