# Reduce this matrix to row-echelon form

I have the following augmented matrix, which I assembled from a system of linear equations.

I want to reduce matrix to row-echelon form

$$\left[\begin{array}{rrr|r} 1 & \lambda & -1 & 1\\ 2 & 1 & 2 & 5\lambda + 1 \\ 1 & -1 & 3 & 4\lambda\\ 1 & -2\lambda & 7 & 10\lambda - 1 \end{array}\right]$$

How should I go about reducing this matrix?

My attempts have yielded matrices such as:

$$\left[\begin{array}{rrr|r} 1 & -1 & 3 & 4\lambda\\ 0 & -1 & -4 & 1-3\lambda \\ 0 & \lambda & -8 & 2- 7\lambda\\ 0 & 0 & -8 & 4-5\lambda \end{array}\right]$$

And

$$\left[\begin{array}{rrr|r} 1 & \lambda & -1 & 1\\ 0 & 3 & -4 & 1-3\lambda \\ 1 & 0 & -9 & 3- 5\lambda\\ 0 & 0 & 32 & 25\lambda - 8 \end{array}\right]$$

But I couldn't zero out one of the rows.

• Subtract the first row from the third? – John Doe May 3 '17 at 15:17
• @JohnDoe The first matrix or the second? – Cobbles May 3 '17 at 15:20
• The second one, so that the 1 in the $(3,1)$ entry gets cancelled out. Is this what you're trying to do? – John Doe May 3 '17 at 15:21
• @JohnDoe I am trying to use gaussian elimination to create an upper triangular matrix – Cobbles May 3 '17 at 15:22
• And I need one of the rows to be zeroes – Cobbles May 3 '17 at 15:23