[Math] Finding a solution when the determinant is zero

linear algebramatricesmatrix equations

I have the set of equations $$x+y-z=-5$$ $$x-y-z=-3$$ $$x+y-z=0$$ I'd just like to point out here that equation one and three are basically the same – just set equal to two different numbers. I don't know if that automatically means there are no solutions or not. Then, I put this into matrix form and got $$\begin{bmatrix} 1&1&-1\\1&-1&-1\\1&1&-1 \end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}-5\\-3\\0\end{bmatrix}$$ (I'll refer to that first matrix as $A$.) I found the determinant of $A$ and it is equal to zero, meaning there is no inverse. I got really stuck here.

I know that to solve this has to do with row echelon form, null space, and column space. I put $A$ into row echelon form and got $$\begin{bmatrix}1&0&-1\\0&1&0\\0&0&0\end{bmatrix}$$ but translating this back out gave $$x-z=-5$$ $$y=-3$$ $$0=0$$ Trying to solve the first equation gave conflicting answers.

So, I guess I have three main questions here:

  1. How do you find the solution here (or figure out that there isn't a solution)?
  2. How do you calculate null space/column space? (Please note I've googled this and I just can't get it.)
  3. How does null space/column space relate to finding the solution here, if at all?

Thanks! Please keep in mind that I'm in 8th grade trying to figure this out, so the simpler the answer, the better. I am willing to try to figure out stuff, though. Any answers would be appreciated.

Edit: Yes, this particular system is unsolvable (thanks to Jack D'Aurizio and others), but I kind of wanted to know how to find a general way to calculate solutions or the absence of solutions when the determinant is zero and the matrix has no inverse. In other words, mainly questions two and three now.

Best Answer

In $Ax=b$ form, the will be at least one solution if and only if $b$ is in the column space of $A$.

If $A$ is a square matrix, there is a unique solution if and only if $\det(A) \ne 0$.

Putting these tests together we have for all square matrices $A$, $Ax=b$ has

  • no solution if $b$ is not in the column space of $A$.
  • a unique solution if $\det(A) \ne 0$.
  • infinitely many solutions if $b$ is in the column space of $A$ but $\det(A) = 0$.

A more general but also slightly more tedious (sometimes) method that will work for non-square matrices is to row reduce the augmented matrix $[A\mid b]$. If you get

  • a row with $[0\mid a]$ where the $0$ represents a row of zeros and $a\ne 0$, then there is no solution.
  • a pivot in each column and no rows with $[0\mid a]$ where $a\ne 0$, then there is a unique solution.
  • at least one column without a pivot and no rows with $[0\mid a]$ where $a\ne 0$, then there are infinitely many solutions.
Related Question