Matrix Equation AXB=C solve for X

[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}

 

[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 …

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

 

[Sage Sky]

thx got it except its goes to new line on matrixs

 

[tiny-tim]

Hi Sage Sky!

… 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 $X=\frac{1}{34}\begin{pmatrix} a & b \\ c & d\end{pmatrix}$

(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)

 

[Borek]

thx got it except its goes to new line on matrixs

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.

 

[Sage Sky]

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

 

[Mark44]

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

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.