1. Simultaneous equations to matrix

So I'm given this question.

$\displaystyle 4x + 2y + 4x = 3, 2y + z = 1, 2x + 2z = 1$

and it says Forumlate a single vector matrix equation from the three equations given.

Sounds pretty obvious but a single vector matrix would just be

|x| = |3|
|y| = |1|
|z| = |1|

Sorry I can't figure out how to set the proper array brackets and stuff. But that is the single matrix vector?

Here is a correct inverse matrix of the matrix you formulated obove.

1 -1 -1.5
0.5 0 -1
-1 1 2

Sorry about the formatting lol. I think I'm missing a step though because I have to show that this matrix is a correct inverse of the matrix I created obove, but the matrix I created was just a single vector??

2. Re: Simultaneous equations to matrix

I don't think this intended a matrix made of a single vector. Rather, they want a single equation (which happens to be a "vector-matrix" equation) rather than three equations.
And that would be
$\displaystyle \begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}$.

3. Re: Simultaneous equations to matrix

Originally Posted by uperkurk
So I'm given this question.

$\displaystyle 4x + 2y + 4z = 3, 2y + z = 1, 2x + 2z = 1$
$\displaystyle M = \left( {\begin{array}{*{20}{c}} 4&2&4\\ 0&2&1\\ 2&0&2\end{array}} \right)$

You want to find $\displaystyle {M^{ - 1}}\left( {\begin{array}{*{20}{c}} 3\\ 1\\ 1 \end{array}} \right)$

4. Re: Simultaneous equations to matrix

Originally Posted by HallsofIvy
I don't think this intended a matrix made of a single vector. Rather, they want a single equation (which happens to be a "vector-matrix" equation) rather than three equations.
And that would be
$\displaystyle \begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}$.

oh right yeh that makes a lot more sense now. Thanks.

@Plato, I can verify that the second part of the question is true. It says show that the following matrix is a correct inverse of the matrix obove.

$\displaystyle \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix}$ by using an indentiy matrix $\displaystyle \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$ so multiplying that matrix by the identity matrix should return $\displaystyle \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}$. ?

5. Re: Simultaneous equations to matrix

Originally Posted by uperkurk
@Plato, I can verify that the second part of the question is true. It says show that the following matrix is a correct inverse of the matrix obove.
$\displaystyle \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix}$ by using an indentiy matrix $\displaystyle \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$ so multiplying that matrix by the identity matrix should return $\displaystyle \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}$. ?
Show that if $\displaystyle M^{-1}=\begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix}$ then
$\displaystyle M\cdot M^{-1}=M^{-1}\cdot M=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$

6. Re: Simultaneous equations to matrix

What the freak! I know how to calculate a matrix but for some reason when I'm doing it manually

$\displaystyle \begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix} X \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix}$ but I end up with 8 for the first position wtf...?

4 x 1 + 2 x 0.5 + 4 x -1 = 8? but it should equal 1?

when I use the online calculator it comes out as $\displaystyle \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$

So what is going on?

7. Re: Simultaneous equations to matrix

Originally Posted by uperkurk
What the freak! I know how to calculate a matrix but for some reason when I'm doing it manually

$\displaystyle \begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix} X \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix}$ but I end up with 8 for the first position wtf...?

4 x 1 + 2 x 0.5 + 4 x -1 = 8? but it should equal 1?

when I use the online calculator it comes out as $\displaystyle \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$

So what is going on?
Arithmetic $\displaystyle (4)( 1) + (2)(0.5) + (4)(-1) = (4)+(1)+(-4)=1$

8. Re: Simultaneous equations to matrix

omg I feel utterly retarded now -_-