Linear Algebra Quiz: Area Volume Scaling

A triangle in the Cartesian plane has vertices at $(0,0)$, $(4,0)$, and $(2,3)$. This triangle is transformed by the linear transformation represented by the matrix $A = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}$. What is the area of the transformed triangle?

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation with standard matrix $A$. If $\det(A) = -2$, which statement accurately describes the effect of $T$ on a solid region $S$ in $\mathbb{R}^3$?

The linear transformation $T(\vec{x}) = A\vec{x}$ with $A = \begin{pmatrix} 2 & -5 \\ 3 & k \end{pmatrix}$ maps a non-degenerate parallelogram in $\mathbb{R}^2$ to a line segment. What is the value of $k$?

A solid object is transformed by the linear map represented by the matrix $M = \begin{pmatrix} 2 & 0 & 0 \\ 1 & 1 & 1 \\ 3 & -1 & 2 \end{pmatrix}$. The resulting solid has a volume of 30 cubic units. What was the volume of the original object?

A unit cube, defined by the vectors $\vec{e_1}, \vec{e_2}, \vec{e_3}$ in $\mathbb{R}^3$, has a volume of 1 cubic unit. It is transformed by the linear transformation $T(\vec{x}) = A\vec{x}$, where $A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & -3 & 1 \\ 2 & 1 & 4 \end{pmatrix}$. What is the volume of the resulting parallelepiped?

A polygon with an area of 10 square units undergoes a linear transformation $T_1$ represented by matrix $A = \begin{pmatrix} 3 & 1 \\ 1 & 1 \end{pmatrix}$, followed by a second linear transformation $T_2$ represented by matrix $B = \begin{pmatrix} 2 & 0 \\ -1 & 2 \end{pmatrix}$. What is the area of the final polygon after both transformations are applied?

A parallelogram with an area of 4 square units is transformed by the matrix $A = \begin{pmatrix} k & 3 \\ -2 & 4 \end{pmatrix}$. The area of the transformed parallelogram is 40 square units. Which of the following is a possible value for $k$?

A linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ is defined by a reflection across the line $y=x$ followed by a projection onto the x-axis. What is the area of the image of the unit square (with vertices at (0,0), (1,0), (0,1), and (1,1)) under this transformation?

A linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ is represented by an invertible matrix $A$. The transformation $T$ is found to triple the volume of any solid. What is the volume scaling factor of the transformation represented by the matrix $2A$?

A triangle $PQR$ has vertices $P=(1,1)$, $Q=(5,2)$, and $R=(2,4)$. This triangle is transformed by the matrix $M = \begin{pmatrix} 3 & -1 \\ 2 & 1 \end{pmatrix}$. What is the area of the transformed triangle?

A parallelepiped in $\mathbb{R}^3$ with a volume of 12 is first scaled by a factor of 2 in the z-direction only. Then, it is sheared according to the transformation with matrix $S = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. What is the volume of the final solid?

Consider the transformation that first reflects vectors across the line $y = x$, then scales by factor 3 in the x-direction and factor 2 in the y-direction. If this composite transformation is applied to a triangle with area 5, what is the area of the transformed triangle?

A linear transformation $T$ maps the unit square $[0,1] \times [0,1]$ to a parallelogram with vertices at $(0,0)$, $(3,1)$, $(2,4)$, and $(5,5)$. What is the area of the image of any region under transformation $T$ if the original region has area 7?

A shear transformation in $\mathbb{R}^2$ is followed by a rotation of $90°$ counterclockwise. If the original parallelogram had area 12 and the final parallelogram has area 36, what is the determinant of the shear transformation matrix?

A transformation $T$ in $\mathbb{R}^3$ has the property that it maps every unit cube to a parallelepiped of volume 6. If $T$ is applied to a tetrahedron with volume 20, what is the volume of the resulting solid?

The linear transformation represented by matrix $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ satisfies $\det(B) = -12$. If this transformation is applied three times consecutively to a triangle with area 4, what is the area of the final triangle?

Consider the transformation matrix $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 2 \\ 1 & 0 & 1 \end{pmatrix}$. If a parallelepiped undergoes this transformation followed by the transformation represented by $A^{-1}$, what happens to its volume?

The unit circle in $\mathbb{R}^2$, which has area $\pi$, is transformed by the matrix $A = \begin{pmatrix} 4 & -2 \\ 1 & 3 \end{pmatrix}$. What is the area of the resulting ellipse?