Compute \(\det\left(\begin{pmatrix}4\end{pmatrix}\right)\).

Compute \(\det\left(\begin{pmatrix}2&3\\5&7\end{pmatrix}\right)\).

Compute \(\det\left(\begin{pmatrix}1&-4\\2&3\end{pmatrix}\right)\).

Find all real t such that \(\det\left(\begin{pmatrix}t&1\\3&2\end{pmatrix}\right)\) = 5.

Compute \(\det\left(\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\right)\).

Compute \(\det\left(\begin{pmatrix}2&1&0\\0&3&4\\0&0&-1\end{pmatrix}\right)\).

Compute \(\det\left(\begin{pmatrix}1&2&3\\0&4&5\\0&0&6\end{pmatrix}\right)\).

Compute \(\det\left(\begin{pmatrix}1&2&3\\0&1&4\\5&6&0\end{pmatrix}\right)\) using expansion along the first row.

Compute \(\det\left(\begin{pmatrix}2&0&1\\3&0&4\\5&1&6\end{pmatrix}\right)\) by expanding along the second column.

A learner computes \(\det\left(\begin{pmatrix}1&2\\3&4\end{pmatrix}\right)\) as \(1\cdot4\) + \(2\cdot3\) = 10. Diagnose the error and give the correct determinant.