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

[mathmathmad]

Homework Statement

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?

Homework Equations

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

The Attempt at a Solution

 

[statdad]

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?

 

[mathmathmad]

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

 

[VeeEight]

Yes, A²=AxA

 

[HallsofIvy]

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

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.

 

[mathmathmad]

ahh I see :)
I canceled 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?)