Find all the matricies so that A*X = X*A

[fluid coral]

I know i have to set up a systems of equations , not sure how to

[carmine marsh]

Multiply XA and AX and equate

What do you get?
$$XA = \begin{matrix}
a & b\
c & d
\end{matrix}
\begin{matrix}
1 &0\
-1 & 1\
\end{matrix}$$

[fluid coral]

I multiplied them , these are the matricies that follow

[carmine marsh]

So
$$\begin{pmatrix}
a-b & b\
c-d & d
\end{pmatrix}$$

and

$$\begin{pmatrix}
a & b\
c-a & d-b
\end{pmatrix}$$

[carmine marsh]

what is up with my latex

[fluid coral]

the first one i got like that

the second one i have in the down left corner -c+d

which is wrong now that i look at it

[carmine marsh]

some of these are unclear make sure you get what i get

[carmine marsh]

redo

[fluid coral]

i got them now

[carmine marsh]

alr very good

[fluid coral]

am i supposed to do smth like a+b = a-b+b? as the first equation

if it works b is 0

[carmine marsh]

one sec
$$a-b = a$$
$$b=b$$
$$c-d=c-a$$
$$d = d-b$$

[carmine marsh]

we're going to move all of them to the left side and make a matrix afterwards

what do you get

[carmine marsh]

-b right

[fluid coral]

this is the first thing i did

does it work like that

or should i only look at a = a-b without combining the expressions

[carmine marsh]

that works too, but we want to deal with all 4 of the equations at once and put them in matrix form

[carmine marsh]

.

[fluid coral]

i then get -a+d = 0

from the third equation

[carmine marsh]

good

and the fourth

[fluid coral]

also d = d because we said that b is equal to 0

[carmine marsh]

moving the terms on the RHS to the LHS?

[fluid coral]

we have d = d-b right? we said that b = 0 , so d = d, if bring d to the left i get 0 = 0

[carmine marsh]

before we begin solving for those, i want use to make all those 4 equations in a matrix form, then we identify free variables etc

[fluid coral]

so i get d-b-d = 0

[carmine marsh]

which is just -b=0 or b=0

good

now give me the augmented matrix

the reason im going this lengthy path is to show you how you would solve parametric matrix equations

[fluid coral]

should i use abcd as the coefficient matrix?

[fluid coral]

i see

[carmine marsh]

we'll use the 4 equations we just simplified (by moving the rhs to the left side)

[fluid coral]

so we have

[carmine marsh]

these

[fluid coral]

wait i'll send you a pic

[carmine marsh]

sure

[fluid coral]

i should use only one solution for the augmented matrix? like i would have for example my equation for a my equation for b and a 0

or two zeros?

[carmine marsh]

the entries of the matrix would simply be the coefficients of a,b,c,d

so if i had equations like

3x + 4y + z = 0
2x + 5y + 2z = 0

the augmented matrix would be like

3 4 1 | 0
2 5 2| 0

[carmine marsh]

but here

we simplified to

b=0
0=0
a-d=0
b=0

lmk if something is not clear

but on second thoughts my method might be too complicated for this

we might as well go with what we are given

a=d, b=0, c= anything [this anything part is what i wanted to show], d=a

[carmine marsh]

,rrcw

,rccw

[fluid coral]

I'm invested now

[carmine marsh]

good ordering, but when writting up the augmented matrix we take the coefficients as the entries not the variables themselves

[fluid coral]

so 1?

[carmine marsh]

whats the coefficient for b

right good

[carmine marsh]

,rccw

[carmine marsh]

okay have you been introduced to the concept of free variables

or row reduction in general

[fluid coral]

Yes

i think atleast my course is in german

[carmine marsh]

$$\begin{pmatrix}
0 & 1 & 0 & 0 & 0\
0 & 0 & 0 & 0 & 0\
1 & 0 & 0 & -1 & 0\
0 & 1 & 0 & 0 & 0
\end{pmatrix}$$

[carmine marsh]

o what do u call free variables in german

[fluid coral]

freie variabeln

[carmine marsh]

nic

[carmine marsh]

also do u notice something about this matrix

[fluid coral]

one row is irrelevant

i mean it's full of 0

[carmine marsh]

right

[fluid coral]

so we can just cancel it

oh i see what you mean

for c

the coefficient is always 0

[carmine marsh]

which is the last row right when we do
$$\begin{pmatrix}
0 & 1 & 0 & 0 & 0\
0 & 0 & 0 & 0 & 0\
1 & 0 & 0 & -1 & 0\
0 & 0 & 0 & 0 & 0
\end{pmatrix}$$

[fluid coral]

so we can set anything

[carmine marsh]

the irrelevant row

[carmine marsh]

can u identify the free variable(s) for me

[fluid coral]

i would suggest the -1

[carmine marsh]

u were close

[carmine marsh]

the variable

[fluid coral]

b is well defined , a = d and c is irrelevant, so it's a and d

we can choose the value for a , then d becomes equal to that and the equation works

or the other way around

we can choose d and a is equal to that and the equation works

[carmine marsh]

how would you define a free variable

i define it as the non-pivot one

[fluid coral]

a variable which i can decide

[fluid coral]

what do you mean by that , not familiar with the word pivot

[carmine marsh]

when doing row reduction, you pick what we call a "pivot"

what are your steps when row reducing

Why would pivot be involved

[carmine marsh]

thats how you row reduce in the first place

[fluid coral]

depends on what matrix i have , but if i have let's say 3 equations and 4 uknowns i would choose one variable arbitrarly and from there i would try to get a matrix where there are only 0 on the left side of the diagonal

[carmine marsh]

lets have a baby matrix

2 1
1 1

[fluid coral]

if i get a row with only 0 i wipe it away

so here i would to I - II

for the first row

1 0

for the second row

2II-I

[carmine marsh]

right so that 1 would be your pivot, the leading entry

[carmine marsh]

actually lemme reorder this for clarity

few row swaps
$$\begin{pmatrix}
1 & 0 & 0 & -1 & 0\
0 & 1 & 0 & 0 & 0\
0 & 0 & 0 & 0 & 0\
0 & 0 & 0 & 0 & 0
\end{pmatrix}$$

[fluid coral]

so what would i be trying to do here? there is no way for me to set the 1s or the -1 to 0

[carmine marsh]

no need to do this, i just do this to see my pivots clearer, can you tell me which columns have leading entries

[fluid coral]

define a leading entry for me

[carmine marsh]

the first non-zero entry in that row

[fluid coral]

it would be the 1 in the first column and row

[carmine marsh]

for example col 1 has a leading entry which is 1

any other columns?

[fluid coral]

col 2 with the 1

[carmine marsh]

good

any others

[fluid coral]

if i look at row wise no , but column wise there is the -1

[carmine marsh]

row wise

so only 1 and 2

[fluid coral]

then no

[carmine marsh]

those then are your pivot columns

[fluid coral]

so what do i do with them?

[carmine marsh]

the remaining others would be your free variables, and we know that we can let the free variables be anything so let c=s and d=t, s and t are real numbers

do you follow

[fluid coral]

and then a = t

and it's solved

no?

[carmine marsh]

b=?
c=?
d=?

[fluid coral]

b = 0

[carmine marsh]

after that we're done

[fluid coral]

c can be whatever

[carmine marsh]

and what did we say that "whatever" is

[fluid coral]

and d is whatever we set it to

[fluid coral]

anything and everything?

[carmine marsh]

.

[fluid coral]

any real number?

[carmine marsh]

a = t, b=0, c=s, d=t s and t are real numbers

yh im just making u repeat

[fluid coral]

aaaaaaaah

[fluid coral]

that last bit was the missing thing

[carmine marsh]

we're done

[fluid coral]

thanks a lot man

[carmine marsh]

whats da matrix

sorry for taking this long

[fluid coral]

the final one you mean

[carmine marsh]

yh yh

[fluid coral]

its

t 0

s t

[carmine marsh]

[fluid coral]

t and s being real numbers

thanksss 🙂

[carmine marsh]

$$\begin{pmatrix}
t & 0\
s & t\
\end{pmatrix}, s, t \in \mathbb{R}$$

[fluid coral]

you explained it very well , better the longer way than the short answer😁

[carmine marsh]

glad i could help

[carmine marsh]

if no other questions i can mark this as solved

[fluid coral]

everything clear, thanks again!

[carmine marsh]

.solved