Conics question

[GeorgeJBrooks]

I came across this question on Conics and I literally have no clue where to start, if someone could help that'd be great!

The matrix form of a polynomial can be written as xTA x + B x + f, where xT = (x,y), A is a 2x2 matrix and B is a 1x2 matrix. Determine the coefficient matrix A for the following polynomial\; -4 x2 -16 xy + 3 y2 + 2 x + 6 y - 45.​

 

[Deleted member 4993]

I came across this question on Conics and I literally have no clue where to start, if someone could help that'd be great!

The matrix form of a polynomial can be written as xTA x + B x + f, where xT = (x,y), A is a 2x2 matrix and B is a 1x2 matrix. Determine the coefficient matrix A for the following polynomial\; -4 x2 -16 xy + 3 y2 + 2 x + 6 y - 45.​

What are your thoughts?

Please share your work with us ...

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

 

[GeorgeJBrooks]

What are your thoughts?

Please share your work with us ...

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

I am stuck right at the beginning

 

[Ishuda]

I came across this question on Conics and I literally have no clue where to start, if someone could help that'd be great!

The matrix form of a polynomial can be written as xTA x + B x + f, where xT = (x,y), A is a 2x2 matrix and B is a 1x2 matrix. Determine the coefficient matrix A for the following polynomial\; -4 x2 -16 xy + 3 y2 + 2 x + 6 y - 45. ​

Please note that it should be "The matrix form of a general quadratic polynomial can be written as ..."

 

[Deleted member 4993]

I am stuck right at the beginning

(x  y) [a  b] (x)
       [c  d] (y)

Can you perform the matrix multiplication above?

 

[GeorgeJBrooks]

(x  y) [a  b] (x)
       [c  d] (y)

Can you perform the matrix multiplication above?

ax+by=0
cx+dy=0

Not sure if this is correct.

 

[stapel]

ax+by=0
cx+dy=0

Not sure if this is correct.

Where is the "equals zero" coming from? Don't create new equations; just do the multiplication to come up with the new expressions.

 

[GeorgeJBrooks]

Where is the "equals zero" coming from? Don't create new equations; just do the multiplication to come up with the new expressions.

So I just get:
[ax+by]
[cx+dy] ? Think the zeros are because they're what i was doing on the precious questions (Sorry for slow replies have been working all day the last couple of days)

 

[Ishuda]

So I just get:
[ax+by]
[cx+dy] ? Think the zeros are because they're what i was doing on the precious questions (Sorry for slow replies have been working all day the last couple of days)

That is just the second two parts, i.e. you have done
[a b] (x)
[c d] (y)

You now need to do the second part
(x y) [ax+by]
.......[cx+dy]
where the .... is just to (hopefully) get it spaced correctly

 

[GeorgeJBrooks]

That is just the second two parts, i.e. you have done
[a b] (x)
[c d] (y)

You now need to do the second part
(x y) [ax+by]
.......[cx+dy]
where the .... is just to (hopefully) get it spaced correctly

x(ax+by)+y(cx+dy)

 

[Ishuda]

x(ax+by)+y(cx+dy)

Now multiply that out, collect like terms, add the f and compare it to the equation
-4 x² -16 xy + 3 y² + 2 x + 6 y - 45.

 

[GeorgeJBrooks]

x(ax+by)+y(cx+dy)

Ax^2+Bxy+Cxy +DY^2+Ex+Fy

A=-4 B=-8 C=-8 D=3
So matrix A=
[-4 -8]
[-8 3]

 

[Ishuda]

Ax^2+Bxy+Cxy +DY^2+Ex+Fy

A=-4 B=-8 C=-8 D=3
So matrix A=
[-4 -8]
[-8 3]

Don't forget the B x, which you have given as Ex + F y, and the f, which you haven't put in yet, so you need values for E, F, and f.

 

[GeorgeJBrooks]

Don't forget the B x, which you have given as Ex + F y, and the f, which you haven't put in yet, so you need values for E, F, and f.

E=2 F=6 and f =-45 ?

 

[Deleted member 4993]

I came across this question on Conics and I literally have no clue where to start, if someone could help that'd be great!

The matrix form of a polynomial can be written as xTA x + B x + f, where xT = (x,y), A is a 2x2 matrix and B is a 1x2 matrix. Determine the coefficient matrix A for the following polynomial\; -4 x2 -16 xy + 3 y2 + 2 x + 6 y - 45.​

Now put those values to define A and B and f.

 

[GeorgeJBrooks]

Now put those values to define A and B and f.

A=
[-4 -8]
[-8 3]
B=
[2]
[6]
f=-45

 

[Ishuda]

A=
[-4 -8]
[-8 3]
B=
[2]
[6]
f=-45

Good answer although on going back and reading the original question, you were right and you you could have stopped at just giving A above.

 

[GeorgeJBrooks]

Good answer although on going back and reading the original question, you were right and you you could have stopped at just giving A above.

Brilliant, thanks for all the help!