I need to solve $\displaystyle \begin{array}{ccc} x + y + z = 50 \\ x + 3y + 5z = 100 \\ x + 3y + 10z = 20 \end{array}$ using matrices.

I set up the matrix equation $\displaystyle \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 3 & 5 \\ 1 & 3 & 10 \end{array} \right] \left[\begin{array}{ccc} x \\ y \\ z \end{array}\right] = \left[ \begin{array}{ccc} 50 \\ 100 \\ 20 \end{array}\right]$

I found the inverse of the first matrix to be $\displaystyle \frac{1}{10} \left[\begin{array}{ccc} 15 & -7 & 2 \\ -5 & 9 & -4 \\ 0 & -2 & 2\end{array}\right]$ (verified with my calculator)

But when multiplying $\displaystyle \frac{1}{10} \left[\begin{array}{ccc} 15 & -7 & 2 \\ -5 & 9 & -4 \\ 0 & -2 & 2\end{array}\right]\left[ \begin{array}{ccc} 50 \\ 100 \\ 20 \end{array}\right] $

I get $\displaystyle \left[\begin{array}{ccc}9 \\ 57 \\-16\end{array}\right]$ (again verified by my calculator)

But the solution is $\displaystyle \left[\begin{array}{ccc} x \\ y \\ z \end{array}\right] = \left[\begin{array}{ccc}29 \\ 17 \\ 4\end{array}\right]$

Pardon me if I'm doing this completely wrong. I'm doing this work based off a couple instructional internet videos. I've never taken a linear algebra course.