Determinants and Volumes

Introduction

Parallelograms and Paralellepipeds

Determinants and Volumes

Volumes of Regions

Each pair of vectors below determines a parallelogram in $\mathbb{R}^2$. Which pair determines a parallelogram of area $0$?

A $3\times 3$ matrix $A$ has $\det(A) = -2$, and $T(x) = Ax$ is the corresponding matrix transformation. Which statement about the image of the unit cube is correct?

A $2\times 2$ matrix $A$ has $\det(A) = 7$. What is the area of the parallelogram determined by the columns of $A$? Enter a number.

Find the area of the triangle with vertices $(0,0)$, $(3,1)$ and $(1,4)$. Enter a number or a fraction, for example 3/2.

The polar change of variables is $x = r\cos\theta$, $y = r\sin\theta$, with Jacobian matrix

Enter the determinant of that matrix - the factor by which the change of variables scales area - as an expression in $r$.

In your own words, state the single most important definition or theorem of this section, and explain what it says.