Given augmented matrix [A|0]: ->RREF ->x=?

  • Thread starter RogerDodgr
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In summary, the given augmented matrix [A|0] is reduced to RREF form, which is correct according to the solution manual. The vector x is determined to be |1|0|0|, which means that x_3 and x_2 must both equal 0 and x_1 can be any value. This solution is any multiple of <1, 0, 0> and the solution space is the vector space spanned by <1, 0, 0>. The value of x_1 = 1 because it is a convenient and easy number to use, but x_1 can be any value.
  • #1
RogerDodgr
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Given augmented matrix [A|0]: -->RREF -->x=?

I started with:
|0 1 0|0|
|0-3 1|0|
|0 0 2|0|
--->RREF--->
|0 1 0|0|
|0 0 1|0|
|0 0 0|0|
The RREF is correct according to solution manual, but
the solution manual says that the vector x =
|1|
|0|
|0|

I think I understand why x_3 and x_2 =0, But I don't understand why x_1 =1.
 
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  • #2
After have the "RREF" what do you do with it?

what that tells you is that solving Ax= 0 is the same as solving
[tex]\left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1\\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{c}x \\ y \\ z\end{array}\right]= \left[\begin{array}{c}0 \\ 0 \\ 0\end{array}\right][/tex]

which is the same as solving the equations 0x+ 1y+ 0z= 0, 0x+ 0y+ 1z= 0, 0x+ 0y+ 0z= 0.
Obviously the first equation tells you that y must be 0. The second equation tells you that z= 0. The third equation doesn't tell you anything!

So how do you know that x= 1? You don't! In fact, x can be anything and <x, 0, 0> would still satisfy that equation. The solution is any multiple of <1, 0, 0>. The "solution space" is the vector space spanned by <1, 0, 0>. In fact, you could use <x, 0, 0> for any x but '1' happens to be a nice easy number.
 
  • #3
Thanks for your help HallsofIvy, I misinterpreted the solution.
 

1. What does the "augmented matrix" represent?

The augmented matrix represents a linear system of equations in a compact form, where the coefficients of the variables are organized in a rectangular array with an additional column representing the constants of the equations.

2. What does "RREF" stand for?

RREF stands for "Reduced Row Echelon Form", which is a form of a matrix that is in its simplest and most organized state, making it easier to solve and analyze the system of equations it represents.

3. What is the purpose of finding the RREF of an augmented matrix?

The purpose of finding the RREF of an augmented matrix is to simplify and standardize the system of equations it represents, making it easier to solve and interpret the solutions. It also allows for the identification of special solutions, such as a unique solution, no solution, or infinitely many solutions.

4. How do you find the RREF of an augmented matrix?

To find the RREF of an augmented matrix, you can use the row operations of elementary row operations, which include: interchanging two rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another row. By applying these operations, you can transform the matrix into its RREF form.

5. How do you interpret the solution for "x" from the augmented matrix?

The solution for "x" can be interpreted by looking at the RREF form of the augmented matrix. If there is a row of zeros followed by a nonzero number in the last column, then the system has no solution. If all the rows are zero except for the last row, then the system has infinitely many solutions. If the matrix is in its simplest form with one leading entry per row, then there is a unique solution for "x".

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