Find the area scale factor of the linear map with matrix \(\begin{pmatrix}2&0\\0&3\end{pmatrix}\).

Find the area of the parallelogram spanned by u = \((1, 2)\) and v = \((3, 4)\).

Find the area of the triangle with vertices \((0, 0)\), \((2, 1)\), and \((1, 4)\).

Find the volume scale factor of the linear map with matrix \(\begin{pmatrix}1&0&0\\0&2&0\\0&0&-3\end{pmatrix}\).

Find the volume of the parallelepiped spanned by \((1, 0, 0)\), \((0, 2, 0)\), and \((0, 0, 5)\).

A \(2\times 2\) matrix has determinant -4. What does this say geometrically about areas and orientation?

A linear map in \(\mathbb R^2\) has determinant 0. What does this mean geometrically?

Find the area scale factor of the shear matrix \(\begin{pmatrix}1&3\\0&1\end{pmatrix}\).

The vectors u = \((2, 4)\) and v = \((1, 2)\) span a parallelogram. Find its area and interpret the result.

A learner says determinant equals area, not signed area scale. Diagnose the error using \(\det\left(\begin{pmatrix}0&1\\1&0\end{pmatrix}\right)\).