1. ## Matrices Help

Determine which of the following formulas hold for all invertible nxn matrices A and B?

1. A+B is invertible
2. (I - A)(I + A) = I - A^2
3. A^9 is invertible
4. (A + A^-1)^3 = A^3 + A^-3
5. AB = BA
6. (A+B)^2 = A^2 + B^2 + 2AB

I originally thought that 4 was the only one not applicable, but apparently that's incorrect.

2. Originally Posted by My Little Pony
1. A+B is invertible
This is wrong, can you see why?
2. (I - A)(I + A) = I - A^2
Open, $\displaystyle I^2 - AI + IA - A^2 = I^2 - A + A - A^2 = I - A^2$
3. A^9 is invertible
Hint: the produce of invertible matrices is invertible.
4. (A + A^-1)^3 = A^3 + A^-3
$\displaystyle (A+A^{-1})^3 = A^3 + 3A^2(A^{-1}) + 3 A(A^{-1})^2 + A^{-3}$
5. AB = BA
Try to find a counterexample.
6. (A+B)^2 = A^2 + B^2 + 2AB
Open, [tex]A^2 + B^2 + AB + BA[/mat].
Therefore, you are assuming that $\displaystyle AB=BA$ which is not always true.

3. ## commutable matrices

in reference to the property
AB = BA,
matrices which satisfy this property as known as commutative matrices and definately doesnt not hold for all real n x n matrices A,B, a simple reasoning behind this is to consider an arbitrary element of the product matrices AB and BA,

i.e. the i,j th entry of AB, or (AB)(i,j) = < row i of A, column j of B>
for AB = BA this would have to equal the i,jth entry of BA and
this would have to hold for all i,j
the i,j th entry of BA, or (BA)(i,j) = <row i of B, column j of A>
i.e. for AB(i,j) = BA(i,j)
<row i of A, column j of B> = <row i of B, column j of A>
and this must hold for all i, j = 1,1 ..... n,n
which obviously cannot be stated in general

Examples of commatative matrices for a given matrix A are
(i) The Identity Matrix
(ii) The Zero Matrix
(iii) The Transpose of the Matrix
(iv) The Matrix itself
(v) The Matrix raised to any real integer power

Hope this helps