Matrix

[Anna2230]

Can anyone explain the steps of solving the following system by performing row operation on the augmented matrix as far x-y+3z=12

4x +z=5

x+3y+z=14 Thank you

 

[HallsofIvy]

You cannot "solve" two equations in three unknowns in the usual sense, you know that, don't you. What you can do is solve for two of the unknowns in terms of the other one. The augmented matrix for this problem is $\displaystyle \begin{bmatrix}4 & 0 & 1 & 5 \\ 1 & 3 & 1 & 14 \end{bmatrix}$. We want to get this into the form $\displaystyle \begin{bmatrix}1 & 0 & a & b \\ 0 & 1 & c & d\end{bmatrix}$ which would be equivalent to the equations "x+ az= b" and "y+ cz= d" which we could then write as x= b- az and y= d- cz, solving for x and y in terms of z. To start with, we want a "1" in the upper left corner. If we swap the two rows (that is one of the three kinds of "row operations") we get $\displaystyle \begin{bmatrix} 1 & 3 & 1 & 14 \\ 4 & 0 & 1 & 5\end{bmatrix}$. I did that just to get a "1" in the upper left corner- swapping rows is simpler than multiplying every number in the first row by 1/4 which is another "row operation". Now, we want a "0" in the lower left corner. We get that by using the third "row operation"- subtract four times the first row from the sconde row: $\displaystyle \begin{bmatrix}1 & 3 & 1 & 14 \\ 4- 4 & 0- 12 & 1- 4 & 5- 56\end{bmatrix}= \begin{bmatrix}1 & 2 & 1 & 14 \\ 0 & -12 & -3 & -51 \end{bmatrix}$. Now we want a "1" in the second place of the second row so we divide each number in the second row by -12: $\displaystyle \begin{bmatrix} 1 & 2 & 1 & 14 \\ 0 & 1 & \frac{1}{4} & \frac{17}{2}\end{bmatrix}$. The last thing we need is a "0" just above the "1" in the second row. We get that by subtracting 2 times the second row from the first row. $\displaystyle \begin{bmatrix}1 & 0 & 1-\frac{1}{2} & 14- \frac{17}{2}\\ 0 & 1 & \frac{1}{2} & \frac{17}{2}\end{bmatrix}= \begin{bmatrix}1 & 0 & \frac{1}{2} & \frac{11}{2} \\ \\ 0 & 1 & \frac{1}{2} & \frac{17}{2}\end{bmatrix}$ which is equivalent to the two equations $\displaystyle x+ \frac{1}{2}z= \frac{11}{2}$ and $\displaystyle y+ \frac{1}{2}z= \frac{17}{2}$ so that $\displaystyle x= \frac{11}{2}- \frac{1}{2}z$ and $\displaystyle y= \frac{17}{2}- \frac{1}{2}z$. z can be any number at all and then those equations give the corresponding values for x and y. For example, if z= 1 then $\displaystyle x= \frac{11}{2}- \frac{1}{2}= 5$ and $\displaystyle y= \frac{17}{2}- \frac{1}{2}= 8$. Or, if z= 3, $\displaystyle x= \frac{11}{2}- \frac{3}{2}= 4$ and $\displaystyle y= \frac{17}{2}- \frac{3}{2}= 7$.

 

[mmm4444bot]

It looks like HallsofIvy may have missed the third equation.

Are you able to write the augmented coefficient matrix, for the system of three equations?

Please show us what you get for that.

explain the steps

I'm not sure that I understand this request.

The steps are known as "Gaussian Elimination"; they convert the augmented coefficient matrix into what's called "reduced row-echelon form", from which you may read the solution for x,y,z.

Have you seen any lessons on "Gaussian Elimination" or "reduced row-echelon form"? What part in the lesson would you like explained?

If you have not worked any exercises like this before, we can look up some examples for you to study. Otherwise, please show whatever work that you can. Cheers :cool:

 

[mmm4444bot]

Please check the summary of our posting guidelines, when you have a chance. Thanks!