Hi guys,
Could i have a little help with the following problems?
Use matrix algebra to solve the following simultaneous equations:
Problem 1:
5x + y = 13
3x +2y = 5
Problem 2:
3x + 2y = -2
x + 4y = 6
I'll step through #1, you try #2 using the same technique.
Set it up like this:
$\displaystyle \begin{bmatrix}5&1&13\\3&2&5\end{bmatrix}$
1/5 times row 1:
$\displaystyle \begin{bmatrix}1&1/5&13/5\\3&2&5\end{bmatrix}$
-3 times row 1 add to row 2, replace row 2 with the result:
$\displaystyle \begin{bmatrix}1&1/5&13/5\\0&7/5&-14/5\end{bmatrix}$
5/7 times row 2:
$\displaystyle \begin{bmatrix}1&1/5&13/5\\0&1&-2\end{bmatrix}$
-1/5 times row 2 add to row 1, replace row 1 with the result:
$\displaystyle \begin{bmatrix}1&0&3\\0&1&-2\end{bmatrix}$
x=3 and y=-2
See what you have to do?. Whittle away until you get it in what is known as reduced row-echelon form. Your solutions are in the far right column and you have 0's and 1's in the other 2 columns.
Because that's what I used to get it into the form I wanted. It didn't have to be those particular numbers. Whatever works. Your method may be different.
You must have some clue of what you must do?. If you don't, you ought to see your teacher. Unless, let me guess, an on-line class.
You have to get the diagonal of 1's and 0's like I did on the 1st problem.
Give it a try with #2 and let's see how you done .
It's a matter of practice and observation.
Hello, c00ky!
Here is another method . . .
Given the matrix equation: .$\displaystyle A X \:=\:C$
. . find $\displaystyle A^{-1}$, the inverse of $\displaystyle A$, and multiply both sides.
We have: .$\displaystyle A^{-1}(A X) \;=\;A^{-1} C\quad\Rightarrow\quad(A^{-1}A)X\:=$ $\displaystyle \:A^{-1}C\quad\Rightarrow\quad IX\:=\:A^{-1}C$
. . Therefore, the solution is: .$\displaystyle X\:=\:A^{-1}C$
It helps if you know this handy formula:
. . The inverse of $\displaystyle A\:=\:\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is: .$\displaystyle A^{-1}\:=\:\frac{1}{ad-bc}\begin{pmatrix}d & -b\\-c & a\end{pmatrix}$
This is easily memorized:
. . (1) Switch on the main diagonal $\displaystyle (\searrow)$
. . (2) Change signs on the other diagonal $\displaystyle (\nearrow)$
. . (3) Divide by the determinant of $\displaystyle A.$
Use matrix algebra to solve the following system:
$\displaystyle 1)\;\begin{array}{cc}5x + y \,= \,13 \\3x +2y \,= \,5\end{array}$
We have: .$\displaystyle \begin{pmatrix}5 & 1\\3 & 2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}\;= \;\begin{pmatrix}13\\5\end{pmatrix}$
. . Then: .$\displaystyle A^{-1} \:=\:\frac{1}{7}\begin{pmatrix}2 & \text{-}1 \\ \text{-}3 & 5\end{pmatrix}$
Hence: .$\displaystyle \begin{pmatrix}x\\y\end{pmatrix} \:= \:\frac{1}{7}\begin{pmatrix}2&\text{-}1\\\text{-}3&5\end{pmatrix}\begin{pmatrix}13\\5 \end{pmatrix} $ $\displaystyle = \:\frac{1}{7}\begin{pmatrix}26-5\\\text{-}39 + 25\end{pmatrix} \;=\;\frac{1}{7}\begin{pmatrix}21\\ \text{-}14\end{pmatrix}
$
. . Therefore: /$\displaystyle \begin{pmatrix}x\\y\end{pmatrix}\;=\;\begin {pmatrix}3 \\ \text{-}2\end{pmatrix}$