1. ## matrix problem

Help!

Solve AX=B, where

⎡ 1 3 1 ⎤ =A
⎣ 2 0 5 ⎦

and

⎡ 1 2 ⎤ =B
⎣ 5 3 ⎦

2. Originally Posted by tohman
Help!

Solve AX=B, where

⎡ 1 3 1 ⎤ =A
⎣ 2 0 5 ⎦

and

⎡ 1 2 ⎤ =B
⎣ 5 3 ⎦
If you mean $A= \begin{bmatrix}1 & 3 & 1 \\ 2 & 0 & 5\end{bmatrix}$ and $B= \begin{bmatrix} 1 & 2 \\ 5 & 3\end{bmatrix}$
(Click on those formulas to see the LaTex code used.)

then "Ax= B" is
$A= \begin{bmatrix}1 & 3 & 1 \\ 2 & 0 & 5\end{bmatrix}\begin{bmatrix} u & v \\ w & x \\ y & z\end{bmatrix}= \begin{bmatrix}1 & 2 \\ 5 & 3\end{bmatrix}$.
Since that maps a 2 by 3 matrix into a 2 by 2 matrix, there will not be a unique answer. The space of 2 by 3 matrices has dimension 6 while the set of 2 by 2 matrices has dimension 4 so you can expect the solution set to have dimension 6- 4= 2 and so depend on two undetermined constants.

There are a number of different ways to solve matrix equations and, if you have been given this problem by a teacher, you are expected to have already seen at least one method. Since, unfortunately, you chose not to show any work at all, we have no way of knowing what method would be appropriate. The two methods most commonly used are:
Write the two given matrices side by side:
$A= \begin{bmatrix}1 & 3 & 1 \\ 2 & 0 & 5\end{bmatrix}\begin{bmatrix}1 & 2 \\ 5 & 3\end{bmatrix}$ and "row reduce" the left matrix as much as possible.

The other, less "sophisticated", method is to write these as four equations, u+ 3w+ y= 1, v+ 3x+ z= 2, 2u+ 5y= 5, and 2v+ 5z= 3. Solve those four equations for four of the unknown values, in terms of the other two.

3. thx a lot! I alreday solved the solutions earlier but I'll solve your equations too, so I can compare the results!