Matrix Problem (Similar)

[kennyken]

This is a similar problem but it should be easier than the one below..

Show that the statement is generally true or find specific 2 x 2 matrices for which the statement is not true.

i) A + B = 0 if and only if A = -B
ii)The matrix equation X^2=I is satisfied only when X=I or X=-I
iii)(A - B)(A+B)=A^2 - B^2 if and only if AB=BA
iv)If B = A^2-5A +2I, then AB+BA
v)AB=O if and only if BA=0
vi)If A^3-7A^2+5I=0, then A^4=49A^2-5A-35I

Points to remember:
I is the identity matrix

 

[soroban]

Hello, kennyken!

I have a few of them . . .

Show that the statement is generally true
or find specific 2 x 2 matrices for which the statement is not true.

ii) The matrix equation X² = I is satisfied only when X = I or X = -I . . . false

. . . [ 2 .1 ]
. . . [-3 -2 ] . satisfies the equation.

iii)(A - B)(A + B) = A² - B² if and only if AB = BA . . . true

(A - B)(A + B) . = . (A - B)A + (A - B)B. . . Distributive property
. . . . . . . . . . . .= . A² - BA + AB - B². . . . Distributive property

If this to equal A² - B², then -BA + AB must equal 0.
. . . Therefore: . AB .= .-BA

v) AB = 0 if and only if BA = 0 . . . false

[2 1] [ 1 .1] . = . [0 0]
[2 1] [-2 -2] . . . .[0 0]

[ 1 .1] [2 1] . = . [ .4 .2]
[-2 -2] [2 1] . . . .[-8 -4]

 

[kennyken]

Thank you Soroban!

Now I only need iv and vi..

 

[tkhunny]

iv)If B = A^2-5A +2I, then AB+BA
vi)If A^3-7A^2+5I=0, then A^4=49A^2-5A-35I

iv appears to be ill-defined. AB+BA = ????

vi

A^3 - 7A^2 + 5I = 0 ==> A^3 = 7A^2 - 5I

Multiply by 'A'
A^4 = 7A^3 - 5A

Substitute for A^3
A^4 = 7(7A^2 - 5I) - 5A

The rest is trivial.

 

[Gene]

I've forgotten more than I remember but I'm dubious about ii) * v)
[ 2 1]
[-3 -2] gives an identity
[1 0]
[0 1] under reduction
and
[ 4 2]
[-8 -4] gives
[1 .5]
[0 0] under reduction.
Aren't they 1 and zero respectively?
I have no opinion as to the true-false answer, just the counter examples.
------------------
Gene

 

[kennyken]

Sorry to tkhunny
The question for iv) is acutally AB=BA

And to Gene. I don't quite understand what you are asking...