I'm trying to solve this problem, which basically states this:
Based on the system of equations of the variables $x, y$ and $z$: $$\begin{cases} x y - 2\sqrt{y} + 3 y z = 8 \\ 2 x y - 3\sqrt{y} + 2 y z = 7 \\ - x y + \sqrt{y} + 2 y z = 4 \\ \end{cases}$$please tell which of the next alternatives are correct:
- The system is possible and determined
- The range of the extended matrix of the system is 2.
- On the transposed matrix of the coefficients, associated with the system: $a_{12} = -3$
- The coefficient matrix associated to the system is non invertible.
So, what I did was construct the extended matrix from the equation system given in the problem statement, replacing the variables as this:
- $a = xy$
- $b = \sqrt{y}$
- $c = yz$
The matrix constructed is this: \begin{bmatrix} 1 & -2 & 3 & 8 \\ 2 & -3 & 2 & 7 \\ -1 & 1 & 2 & 4 \end{bmatrix} Solving by gaussian method, the matrix is reduced to: \begin{bmatrix} 1 & -2 & 3 & 8 \\ 0 & 1 & -4 & -9 \\ 0 & 0 & 1 & 3 \end{bmatrix}
So finally:
- $a = 5$
- $b = 3$
- $c = 3$
The values of $x, y$ and $z$ could be calculated using the previous equations, giving this values as results:
- $x = 5/9$
- $y = 9$
- $z = 1/3$
This is the far as I can go on solving this problem, my question basically is, how to transform this to a linear equation system of x, y and z, because I do have the values of $x, y$ and $z$ but the statements that I have to indicate if they are correct or not, are based on that matrix that I don't know how to build.
I was thinking that maybe this problem doesn't look for me to build such a matrix, instead, given that there exist a solution for the system proposed initially, I should deduct the veracity of the statements:
- First statement is correct as each variable has just one unique value.
- Second statement is incorrect because for each variable having one single value, the range of the matrix should be 3.
Second and third statement I have no idea how to answer to.
Please give me a hand on this. Thank you a lot.