Simultaneous equations to matrix

**uperkurk** · August 23rd 2012, 09:13 AM

So I'm given this question.

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

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??

**HallsofIvy** · August 23rd 2012, 09:23 AM

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
$\begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}$ .

**Plato** · August 23rd 2012, 09:32 AM

Originally Posted by uperkurk

So I'm given this question.

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

$M = \left( {\begin{array}{*{20}{c}} 4&2&4\\ 0&2&1\\ 2&0&2\end{array}} \right)$

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

**uperkurk** · August 23rd 2012, 10:09 AM

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
$\begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}$ .

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.

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

**Plato** · August 23rd 2012, 10:29 AM

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

Show that if $M^{-1}=\begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix}$ then
$M\cdot M^{-1}=M^{-1}\cdot M=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$

**uperkurk** · August 23rd 2012, 12:22 PM

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

$\begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix} X \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix}$ 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 $\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$

So what is going on?

**Plato** · August 23rd 2012, 12:26 PM

Originally Posted by uperkurk

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

$\begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix} X \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix}$ 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 $\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$

So what is going on?

Arithmetic $(4)( 1) + (2)(0.5) + (4)(-1) = (4)+(1)+(-4)=1$

**uperkurk** · August 23rd 2012, 12:43 PM

omg I feel utterly retarded now -_-

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