Apply \(\mathbb R^2\) <- \(\mathbb R^2\) - 3R1 to \(\begin{pmatrix}1&2&5\\3&7&20\end{pmatrix}\).

Apply \(\mathbb R^1\) <-> \(\mathbb R^2\) to \(\begin{pmatrix}0&2&4\\1&-1&3\end{pmatrix}\).

Apply \(\mathbb R^2\) <- (1/4)\(\mathbb R^2\) to \(\begin{pmatrix}1&0&2\\0&4&8\end{pmatrix}\).

Apply \(\mathbb R^3\) <- \(\mathbb R^3\) + 2R1 to \(\begin{pmatrix}1&-1&0\\0&1&3\\-2&4&5\end{pmatrix}\).

Which elementary row operation would turn \(\begin{pmatrix}2&1&5\\6&4&17\end{pmatrix}\) into \(\begin{pmatrix}2&1&5\\0&1&2\end{pmatrix}\)?

Apply the two operations \(\mathbb R^2\) <- \(\mathbb R^2\) - 2R1, then \(\mathbb R^3\) <- \(\mathbb R^3\) + \(\mathbb R^1\) to \(\begin{pmatrix}1&2&1\\2&5&4\\-1&0&3\end{pmatrix}\).

Explain why multiplying one row by 0 is not an elementary row operation.

Apply \(\mathbb R^1\) <- \(\mathbb R^1\) + 5R2 to \(\begin{pmatrix}1&-5&7\\0&1&-2\end{pmatrix}\).

Find k so that applying \(\mathbb R^2\) <- \(\mathbb R^2\) + kR1 to \(\begin{pmatrix}2&1&0\\6&5&1\end{pmatrix}\) makes the first entry of \(\mathbb R^2\) equal to 0.

A learner changes \(\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}\) to \(\begin{pmatrix}1&2&3\\4&5&0\end{pmatrix}\) by altering only the final entry. Explain why this is not a valid row operation.