# Matrix Equation AXB=C solve for X

1. Jun 29, 2012

### Sage Sky

Hi

I have a question in my math coursework on matrices
Question is
Three matrices A, B and C are given by:

$$A = \begin{pmatrix} 2 & 5 \\ -4 & 1 \end{pmatrix}$$
$$B = \begin{pmatrix} 1 & 3 \\ -2 & 6 \end{pmatrix}$$
$$C = \begin{pmatrix} 2 & 1 \\ -3 & -1 \end{pmatrix}$$

a) find the inverses of A and B
b) Solve the equation AXB=C

This is my solution

iknow have rearranged equation to make it

${X=}A\frac{C}{A B}$

i understand that you can't dived in matrices so i have to use the inverse of A and B
to make the equation ${X=}{A}^{-1} {C} {B}^{-1}$

The det $\left|A\right|$ = 22
det $\left|B\right|$ = 12

$A^{-1} = \frac{1}{22}$ \begin{pmatrix} 1 & -5 \\ 4 & 2 \end{pmatrix}
$B^{-1} = \frac{1}{12}$ /begin{pmatrix} 6 & -3 \\ 2 & 1 \end{pmatrix}

and my answer for inv A mutliplied C is \frac{1}{22}[/itex] \begin{pmatrix} 17 & 6 \\ 2 & 2 \end{pmatrix} or \begin{pmatrix} 0.7727 & 0.2727 \\ 0.0909 & 0.0909\end{pmatrix}

and then i mutlipy by inv B i get matrix X= \begin{pmatrix} 0.4318 & -0.1705 \\ 0.0606 & -0.0152\end{pmatrix}

my question is this answer correct and how can i show the answer for Matrix X as $X=\frac{1}{?}$\begin{pmatrix} a & b \\ c & d\end{pmatrix}

do i add the $\frac {1}{22}+ \frac{1}{12}$which would make it $\frac{1}{34} ??????$

would this then make Matrix X $X=\frac{1}{34}$\begin{pmatrix} a & b \\ c & d\end{pmatrix}

Last edited by a moderator: Jun 29, 2012
2. Jun 29, 2012

### tiny-tim

Welcome to PF!

Hi Sage Sky! Welcome to PF!

If you're having trouble writing matrices in latex,

then try the CODE button instead, like this …

Code (Text):

1 2 3    1 3 2
4 5 6    4 6 7
7 8 9    9 8 7

3. Jun 29, 2012

### Sage Sky

thx got it except its goes to new line on matrixs

4. Jun 29, 2012

### tiny-tim

Hi Sage Sky!
(i haven't checked your figures, but the method looks ok)

you could have kept the factors 1/22 and 1/12 outside the matrices all the time

(so you'd be multiplying them in the end)

5. Jun 29, 2012

### Staff: Mentor

It doesn't. You were doing strange tricks to the opening and closing tags (you were opening them and closing them in random places).

And be consistent - don't mix LaTeX with non LaTeX formulas.

6. Jun 29, 2012

### Sage Sky

ok so it would be $\frac{1}{264} \begin{pmatrix} 114 & -45 \\ 16 & -4\end{pmatrix}$

7. Jun 29, 2012

### Staff: Mentor

You can check your answer by multiplying AXB and comparing it to C. Don't bring the fractions into your matrices - you can move them all the way to the left and multiply all of them together and all the matrices together.