# Reducing the Matrix to Reduced Row Echelon Form

Reduce the matrix $\begin{bmatrix}1&-1&-6\\4&-1&-15\\-2&2&12\end{bmatrix}$ to reduced row-echelon form

I performed the row operations:

1) $R_2 = 4R_1 - R_2$

2) $R_3 = 2R_1 + R_3$

3) $R_2 = R_2 / -3$;

4) $R_3 = R_3/18$

5) $R_2 = R_2 + 7R_3$

6) $R_1 = R_1 + -6R_3$

7) $R_1 = R_1 + R_2$

Which gives me the RREF of the matrix

$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

So how in the world is my solution incorrect?

• According to Wolfram Alpha, this has a zero determinant, so the RREF can not be the identity matrix. I don't know exactly where you went wrong, though. – Noble Mushtak Jun 23 '16 at 20:30
• $-2R_1=R_3$ so there's no way this reduces to the identity. – user137731 Jun 23 '16 at 20:30
• I don't either... – Shammy Jun 23 '16 at 20:30
• @Shammy After step 2, you should get that the third row is completely zero. – Noble Mushtak Jun 23 '16 at 20:32
• $R_3 = -2 R_1$. After step 2, Row 3 should be a zero vector. Any steps involving row 3 that follow must be incorrect. – Doug M Jun 23 '16 at 20:32

After Step 1:

$$\left[\begin{matrix}1 & -1 & -6 \\ 0 & -3 & -9 \\ -2 & 2 & 12\end{matrix}\right]$$

After Step 2:

$$\left[\begin{matrix}1 & -1 & -6 \\ 0 & -3 & -9 \\ 0 & 0 & 0\end{matrix}\right]$$

After Step 3:

$$\left[\begin{matrix}1 & -1 & -6 \\ 0 & 1 & 3 \\ 0 & 0 & 0\end{matrix}\right]$$

All of the steps with $R_3$ are unnecessary since $R_3$ is all zeroes. Skip to Step 7:

$$\left[\begin{matrix}1 & 0 & -3 \\ 0 & 1 & 3 \\ 0 & 0 & 0\end{matrix}\right]$$

• -9 in first step?? what??????? how???????? – Shammy Jun 23 '16 at 20:39
• @Shammy $$4\cdot -6-(-15)=-24+15=-9$$ – Noble Mushtak Jun 23 '16 at 20:39
• I'm an idiot. On my paper I had a positive 6 in the third column when I wrote it down instead of a -6 -_-. I hate matrices – Shammy Jun 23 '16 at 20:41
• That's why I like to use Wolfram Alpha to check my work. For example, here's Wolfram Alpha's answer to the RREF of this matrix. – Noble Mushtak Jun 23 '16 at 20:44
• I finally got it. Thank you @noble you truly are, Noble – Shammy Jun 23 '16 at 20:50