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# Matrices watch

1. How would I find

1. A real 2x2 matrix that satisfies where is the identity matrix?

2. 2x2 matrices that satisfy ?
2. 1-Write A as (a b)
(c d).
Solve the simultaneous equations that arise.
3. (Original post by TheEd)
A 2x2 matrix that satisfies where is the identity matrix?
4. (Original post by Pheylan)
I forgot to mention it has to be REAL, not imaginary.
5. (Original post by TheEd)
I forgot to mention it has to be REAL, not imaginary.
You'll discover conditions on the matrix from doing this, which should lead you towards the answer.
6. If we are told

Then we have:

(1)
(2)
(3)
(4)

From (1) and (4) we have
Or,
Then, from (2),

So, for real solutions, and where c does not equal b. That is to say our solutions are:

and

We also have two complex solutions where
7. (Original post by tgodkin)
If we are told

Then we have:

OK cheers I'll have a go at solving them.
8. For your second problem, to get you on the right track:

AB=-BA is saying…

9. (Original post by tgodkin)
If we are told

Then we have:

(1)
(2)
(3)
(4)

From (1) and (4) we have
Or,
Then, from (2),

So, for real solutions, and where c does not equal b. That is to say our solutions are:

and

We also have two complex solutions where
Can you please explain how you arrived at those real solutions from solving the simultaneous equations?
10. (Original post by tgodkin)
If we are told

Then we have:

(1)
(2)
(3)
(4)

From (1) and (4) we have
Or,
Then, from (2),

So, for real solutions, and where c does not equal b. That is to say our solutions are:

and

We also have two complex solutions where
There's an infinite number of solutions with real coefficients, you've missed a fair few.

a^2=d^2 doesn't imply a=d.
11. (Original post by TheEd)
Can you please explain how you arrived at those real solutions from solving the simultaneous equations?
If b or c were to be 0, it is clear from (1) than a is a complex number.
So, let us assume is a solution, then .

Two answers satisfy this equation which are:

where
12. (Original post by Slumpy)
There's an infinite number of solutions with real coefficients, you've missed a fair few.

a^2=d^2 doesn't imply a=d.
True I overlooked that. If we only consider real solutions then could also imply
This however doesn't change the values of our real solutions.

But yes I can see there would be many complex solutions, thanks.
13. (Original post by tgodkin)
True I overlooked that. If we only consider real solutions then could also imply
This however doesn't change the values of our real solutions.

But yes I can see there would be many complex solutions, thanks.
It really does, there are, as I say, an infinite number of them.
Pick a=-d.
Now pick bc such that b=-c, and |bc|=a^2+1.
14. (Original post by Slumpy)
It really does, there are, as I say, an infinite number of them.
Pick a=-d.
Now pick bc such that b=-c, and |bc|=a^2+1.
Yeah I see that now, thanks.
15. (Original post by tgodkin)
For your second problem, to get you on the right track:

AB=-BA is saying…

Well I've found 2 solutions for this question

and

but that was pure guess work! I tried it with simultaneous equations but how can you do it? They're all unknowns.

I got the simultaneous equations down to ae=dh but couldn't get any further.
16. For (2), A = B = zero matrix will work.

Also, it may help to think of (1) geometrically. Let A represent a rotation, for example.

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