Elementary Row Operation/Examples/Operations on Arbitrary Matrix/lambda r2

Example of Elementary Row Operation
Let $\mathbf A$ be the matrix:
 * $\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$

Let the elementary row operation $e$ be applied to $\mathbf A$, where $e$ is defined as:
 * $e := r_2 \to \lambda r_2$

Then $\mathbf A$ is transformed into:
 * $\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 \lambda & -\lambda & \lambda & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$

Proof
From Elementary Row Operation: $r_2 \to \lambda r_2$, the elementary matrix $\mathbf E$ corresponding to $e$ is:
 * $\mathbf E = \begin {pmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & 1 \end {pmatrix}$

Hence by Elementary Row Operations as Matrix Multiplications: