Solving MIT OCW Problem with A Matrix

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In summary, the conversation discusses a problem found on the MIT OpenCourseWare website that involves finding the complete solution to a system involving the original matrix A. The solution involves using the reduced row echelon form of A and setting the equations equal to zero. The general solution is then found by solving for the free variables.
  • #1
amcavoy
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I found this problem on the MIT OpenCourseWare website, but a solution was not given. I tried it out, so I was wondering if my answer is correct. The problem is as follows:

----------------------------------------------------------

Suppose A is reduced by the usual row operations to

[tex]R=\begin{bmatrix}1 & 4 & 0 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0\end{bmatrix}.[/tex]

Find the complete solution (if a solution exists) to this system involving the original A:

Ax = sum of the columns of A.

----------------------------------------------------------

I figured if you took the columns of A and added them together, it would be the same as multiplying a by <1,1,1>. Thus I have the following:

[tex]A\mathbf{x}=A\begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}.[/tex]

So the solution is the vector <1,1,1>. Is this correct?

Thank you.
 
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  • #2
no,the solution is a general solution not a single solution or no solution. depending on what R is R=A or R=A|b(augmented matrix) the solution is
R=A ( id oubt its this one)
x1+4*x2+2*x4=0;
x3+2*x4 = 0

R=A|b
x1+4*x2=2
x3 =2
therefore the solution is ->
x=
|2-4t|
|..t..|
|..2..|

you arelooking at the rows.
[]if there was no solution a row would have all 0s except the last one=#(!=0)
thus 0=# which is false
[]if there was 1 solution than all the xi would be solvable...that is there must be a few rows where its just xi=# like the 2nd row. THen the remaining rows would be solved by the other linear equations in the matrix.
[]if there was many solutions than you'd have hte above.
 
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  • #3
So, for R=A:

[tex]x_1+4x_2+2x_3=0[/tex]

[tex]x_3+2x_4=0[/tex]

For which the general solution is:

[tex]x_2\begin{bmatrix}-4 \\ 1 \\ 0 \\ 0\end{bmatrix}+x_4\begin{bmatrix}4 \\ 0 \\ -2 \\ 1\end{bmatrix}\quad\forall x_2,x_4\in\mathbb{R}[/tex]

I understand how to find the general solution and everything, but what I would like to know is why you set the equations equal to zero. If R=A, then the sum of the columns of A should equal the sum of the columns of R, which is non-zero. Could you just explain your steps when you set R=A and R=[A|b]?

Thanks again.
 
  • #4
if the system is Ax=b; then R=[A|b] where the last column of R is b and each remaining column of R represents a xi; I believe that is the solution

however because you did not state the question the solution could be R=A where you did not state b...and so i assumed b=O
 
  • #5
Alright I understand what you are trying to say.

[tex]A\mathbf{x}=A\mathbf{u}\quad \mathbf{u}=\begin{bmatrix}1 \\ 1 \\ 1 \\ 1\end{bmatrix}[/tex]

[tex]A(\mathbf{x}-\mathbf{u})=0[/tex]

So upon solving, I come up with the general solution which is:

[tex]x_2\begin{bmatrix}-4 \\ 1 \\ 0 \\ 0\end{bmatrix}+x_4\begin{bmatrix}-2 \\ 0 \\ -2 \\ 1\end{bmatrix}+\begin{bmatrix}1 \\ 1 \\ 1 \\ 1\end{bmatrix}\quad\forall x_2,x_4\in\mathbb{R}[/tex]

If I didn't make any arithmetic mistakes.
 

1. What is MIT OCW?

MIT OCW (OpenCourseWare) is a free and open online platform that provides access to course materials from the Massachusetts Institute of Technology (MIT). It offers a wide range of educational resources, including lecture notes, problem sets, and exams.

2. How can solving MIT OCW problems with a matrix be useful?

Solving MIT OCW problems with a matrix can be useful for practicing mathematical and analytical skills, as well as for understanding the application of matrices in real-world problem solving.

3. What is a matrix?

A matrix is a rectangular array of numbers or symbols that can be used to represent data or solve mathematical problems. It consists of rows and columns, and can be manipulated using various operations.

4. Are there any specific guidelines for solving MIT OCW problems with a matrix?

Yes, there are some general guidelines for solving MIT OCW problems with a matrix. These include identifying the given information and variables, setting up a matrix equation, and using appropriate matrix operations to solve for the unknown variables.

5. Can I use any type of matrix to solve MIT OCW problems?

Yes, you can use different types of matrices, such as square matrices, rectangular matrices, or even augmented matrices, to solve MIT OCW problems. The type of matrix used will depend on the specific problem and the operations needed to solve it.

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