# Finding the inverse of a matrix by Gaussian elimination

I spent last hours trying to figure out how to solve the inverse matrix to this matrix: $$\begin{pmatrix} 2 &-3 & 1 \\ 1 & 2 &-1 \\ 2 & 1 & 1 \end{pmatrix}$$

The correct result should be $$\begin{pmatrix} 0.250 & 0.333 & 0.083 \\ -0.250 & 0.000 & 0.250 \\ -0.250 & -0.667 & 0.583 \end{pmatrix}$$

However, I am still unable to get there. Here how I tried it (with using the Gaussian Elimination Rule):

$$\begin{multline} \left( \begin{array}{ccc|ccc} 2 & -3 & 1 & 1 & 0 & 0 \\ 1 & 2 & -1 & 0 & 1 & 0 \\ 2 & 1 & 1 & 0 & 0 & 1 \end{array} \right) \overset{[1] - 2[2] \rightarrow [2]}{\Longrightarrow} \left( \begin{array}{ccc|ccc} 2 & -3 & 1 & 1 & 0 & 0 \\ 0 & -7 & 2 & 1 & -2 & 0 \\ 2 & 1 & 1 & 0 & 0 & 1 \end{array} \right) \overset{[1] - [3] \rightarrow [3]}{\Longrightarrow} \\ \left( \begin{array}{ccc|ccc} 2 & -3 & 1 & 1 & 0 & 0 \\ 0 & -7 & 2 & 1 & -2 & 0 \\ 0 & -4 & 0 & 1 & 0 & -1 \end{array} \right) \overset{4[2] - 7[3] \rightarrow [3]}{\Longrightarrow} \left( \begin{array}{ccc|ccc} 2 & -3 & 1 & 1 & 0 & 0 \\ 0 & -7 & 2 & 1 & -2 & 0 \\ 0 & 0 & 8 & -3 & -8 & 7 \end{array} \right) \overset{4[2] - [3] \rightarrow [2]}{\Longrightarrow} \\ \left( \begin{array}{ccc|ccc} 2 & -3 & 1 & 1 & 0 & 0 \\ 0 & -28 & 0 & 7 & 0 & -7 \\ 0 & 0 & 8 & -3 & -8 & 7 \end{array} \right) \overset{8[1] - 3[2] \rightarrow [1]}{\Longrightarrow} \left( \begin{array}{ccc|ccc} 16 & -24 & 0 & 11 & 8 & -7 \\ 0 & -28 & 0 & 7 & 0 & -7 \\ 0 & 0 & 8 & -3 & -8 & 7 \end{array} \right) \Longrightarrow \\ \left( \begin{array}{ccc|ccc} 2 & -3 & 0 & \tfrac{11}{8} & 1 & \tfrac{-7}{8} \\ 0 & -7 & 0 & \tfrac{7}{4} & 0 & \tfrac{-7}{4} \\ 0 & 0 & 8 & -3 & -8 & 7 \end{array} \right) \overset{7[1] - 3[2] \rightarrow [1]}{\Longrightarrow} \left( \begin{array}{ccc|ccc} 14 & 0 & 0 & \tfrac{35}{8} & 7 & \tfrac{-7}{8} \\ 0 & -7 & 0 & \tfrac{7}{4} & 0 & \tfrac{-7}{4} \\ 0 & 0 & 8 & -3 & -8 & 7 \end{array} \right) \Longrightarrow \\ \left( \begin{array}{ccc|ccc} 1 & 0 & 0 & 0.3125 & 0.5 & -0.0625 \\ 0 & 1 & 0 & -0.25 & 0 & 0.25 \\ 0 & 0 & 1 & -0.375 & -1 & 0.875 \end{array} \right) \end{multline}$$

(Original images: one, two)

I would be very grateful guys for helping me to figure out what I am doing wrong, there's always something why the whole inverse matrix is not correct.

Thank you very much

• alright I've done this on scratch paper and I've revealed the row operations on a few comments below someone's answer. However, I think the number on the top left is wrong... the answer to the entire first row should be $\frac{1}{4}. \frac{1}{3}$ and $\frac{2}{3}$ the answer to the second and third rows are correct May 1, 2014 at 10:14
• I have attempted to improve the readability of your question by introducing MathJax. Please check to ensure that I have not made any errors in transcribing your images. May 2, 2014 at 9:09

• try do these row operations $-r_1+r_3 \rightarrow r_3$........ $-r_1+2r_2 \rightarrow r_2$... sorry I'm doing this in my head.. May 1, 2014 at 9:35
• ok... I'm doing this on paint right now ... try these row operations for now $-r_1+ r_3 \rightarrow r_3$.........$-r_1+2r_2 \rightarrow r_2$...........$r_2 \leftrightarrow r_3$..........$-7r_2+4r_3 \rightarrow r_3$....$\frac{1}{4}r_2 \rightarrow r_2$......$\frac{-1}{12} r_3 \rightarrow r_3$ wait a sec I'm missing something.. umm can you tell me what the right side of the matrix is? I think row operations must also happen as well. so we are doing row operations for those two matrices... appears to be an augmented matrix. A l B May 1, 2014 at 9:39