What will be the formula when ad-bc=0 what will be the formula then?

[mellow acorn]

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[trail garnet]

It won't be a group, as inverses wouldn't exist.

[mellow acorn]

I see

And when it is not equals to 0 what will be the formula

[trail garnet]

Sorry, haven't studied group theory, so I don't know.

[mellow acorn]

It's okay

48 is answer

I am asking two questions

One is related to given question

And second is what if ad-bc=1 then formula will be?

[undone sky]

formula of what

[mellow acorn]

Of that matrix?

[undone sky]

wdym "formula of matrix"?

[mellow acorn]

Modulo 3 (48)

I didn't understand it

[undone sky]

you can count them

[mellow acorn]

Like how?

[undone sky]

1. Verify that G is a group

this condition is likely a typo

[mellow acorn]

Why?

It should be 0?

[undone sky]

ad - bc

i.e the determinant

the general formula for counting 2x2 invertible matrices modulo prime p is (p^2-1)(p^2-p)

plug in p = 3 and you get 48

[mellow acorn]

Yeah 48 true

if we set ad=bc+1

[undone sky]

if you want determinant = 1 specifically consider ad - bc = 1

how do you get A-B = 1 modulo 3?

[mellow acorn]

Is there any specific formula?

[undone sky]

what can A and B be

[mellow acorn]

A=2, B=1 ?

[undone sky]

yes

any other ways?

[mellow acorn]

Can I take any number?

A=4,B=3

[undone sky]

you are calculating in Z3

4=1 and 3=0

[mellow acorn]

Ohh {0,3,6,9,12 and negatives}

We can not get 1 as remainder

[undone sky]

does 0 - 2 work?

[mellow acorn]

2 and 1 are not in 3z?

[undone sky]

Z3 consists of 3 elements (3 equivalence classes): call them 0,1 and 2

[mellow acorn]

It is multiplication?

[undone sky]

both

• and *

[mellow acorn]

Is this wrong?

[undone sky]

these are multiples of 3

i.e 0 in Z3

the set {... -9,-6,-3,0,3,6,9,...}

[mellow acorn]

Then 0-2 works

[undone sky]

is considered as one element

any other ways?

[mellow acorn]

1-0

[undone sky]

correct

[mellow acorn]

0 1 2

1-0,2-1,0-2

3 ways

Ohh wait

[undone sky]

that's right

[mellow acorn]

0-1 wrong

No?

[undone sky]

1-0

[mellow acorn]

Yeah

So total 3 ways?

[undone sky]

so take the case 1-0

how can you make ad = 1 and bc = 0?

[undone sky]

yes

[mellow acorn]

what will be a and d?

[undone sky]

well they have to be inverses of one another dont they?

[mellow acorn]

Yes they have

But 0 isn't

[undone sky]

so it can be 11,22

[mellow acorn]

What?

I didn't understand it

[undone sky]

a=1 d=1 or a=2 d=2

[mellow acorn]

We are discarding

Because a and d have to be inverse of each other?

[undone sky]

discarding what

[mellow acorn]

1-0 option

It can not happen?

[undone sky]

what?

i am asking you to count how many matrices fall in the 1-0 category

[mellow acorn]

Yes

[undone sky]

so how can you make 0?

[mellow acorn]

By multiplying 0?

[undone sky]

bc = 0, what can b and c be?

[mellow acorn]

One of them should be 0

[undone sky]

yes, because z3 is a field

[mellow acorn]

Yes

{0,1,2}

[undone sky]

so 00,01,02,10,20

so for 1-0 case we have

ad in {11,22}

bc in {00,01,02,10,20}

how many such matrices do we get then?

[mellow acorn]

Let me think

ad=1

How you get {11,22}

i meant 1×1 =1

But 2×2=4

[undone sky]

4 = 4-3 = 1

[mellow acorn]

Ohh divided by 3?

[undone sky]

no

[mellow acorn]

1 remainder

[undone sky]

yes

[undone sky]

this set is 0

similarly this set {..,-5,-2,1,4,7,..} is 1

[mellow acorn]

Yeah each one gives 1 remainder when divided by 3

I will back in 5 minutes

Sorry

Now we count

Total 100 matrices?

2×2×5×5?

[undone sky]

you are not making sense

[mellow acorn]

Wait no

[undone sky]

recall that there are 48 invertible matrices

[mellow acorn]

I made mistake it will be just 10

[undone sky]

good

[mellow acorn]

Thank god

Now 2-1

Let me do it now

And you check then

Deal?

[undone sky]

now do the remaining cases

[mellow acorn]

4 cases 2-1

10 cases

0-2

So total 24

@undone sky

Please check

[undone sky]

how do you come up with 4 cases for 2-1?

[mellow acorn]

,rotate

[undone sky]

, rotate

[undone sky]

yes, correct

[mellow acorn]

Thank you☺️

So my solution is completed?

[undone sky]

yes

[mellow acorn]

Can I ask next doubt?

[undone sky]

shoot

[mellow acorn]

7th

[undone sky]

oh no

new topic

close this one

[mellow acorn]

Sure sure

Tq very much

+close