Determinants as Oriented Area and Volume

Definition An ordered pair $(\mathbf{u}, \mathbf{v})$ of linearly independent vectors in $\mathbb{R}^2$ is positively oriented if the counterclockwise angle from $\mathbf{u}$ to $\mathbf{v}$ lies strictly between $0$ and $\pi$, and negatively oriented if it lies strictly between $\pi$ and $2\pi$. The standard pair $(\mathbf{e}_1, \mathbf{e}_2)$, first right and then up, is positively oriented and is the reference orientation of the plane. In $\mathbb{R}^3$ an ordered triple $(\mathbf{u}, \mathbf{v}, \mathbf{w})$ of linearly independent vectors is positively oriented, or right-handed, if $\mathbf{w}$ lies on the side of the plane $\operatorname{span}\{\mathbf{u}, \mathbf{v}\}$ where the right thumb points when the fingers point along $\mathbf{u}$ and curl toward $\mathbf{v}$ through the smaller angle. Otherwise the triple is negatively oriented, or left-handed. The standard triple $(\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3)$ is right-handed. A dependent pair or triple has no orientation.

Definition The oriented area of an ordered pair $(\mathbf{u}, \mathbf{v})$ of vectors in $\mathbb{R}^2$ is the area of the parallelogram

with a plus sign when the pair is positively oriented and a minus sign when it is negatively oriented. When $\mathbf{u}$ and $\mathbf{v}$ are dependent, the parallelogram is a segment or a point and the oriented area is $0$. The determinant of a real $2 \times 2$ matrix $A = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 \end{bmatrix}$ is the oriented area of its ordered columns $(\mathbf{a}_1, \mathbf{a}_2)$. Since $A\mathbf{e}_1 = \mathbf{a}_1$ and $A\mathbf{e}_2 = \mathbf{a}_2$, the image of the unit square is the parallelogram $P(\mathbf{a}_1, \mathbf{a}_2)$, so $\lvert \det A \rvert$ is its area and the sign records the order of its two edges. In the same way, the determinant of a real $3 \times 3$ matrix is the oriented volume of its three ordered columns, the volume of the parallelepiped they span with a plus sign for a right-handed triple, a minus sign for a left-handed one, and $0$ for a dependent one. Only square matrices have determinants. The next lesson defines the determinant of an $n \times n$ matrix for every $n$ and shows how to compute it.

Definition For all real numbers $a, b, c, d$,

To derive it, suppose first that $a \ne 0$. The shear $(x, y) \mapsto (x, y - \tfrac{c}{a}x)$ slides each vertical line along itself, so it keeps areas and does not flip the plane. It sends the first column $(a, c)$ to $(a, 0)$ and the second column $(b, d)$ to $(b, d - \tfrac{bc}{a})$. The first edge now lies on the horizontal axis with signed length $a$, and the second edge has signed height $d - \tfrac{bc}{a}$. The area is base times height, and the pair is positively oriented exactly when the signed length and the signed height have the same sign, so the oriented area is

If $a = 0$ and $c \ne 0$, the first edge $(0, c)$ is vertical. Sliding the second edge parallel to it, from $(b, d)$ to $(b, 0)$, keeps the oriented area, and the pair $((0, c), (b, 0))$ has oriented area $-bc$, which is $ad - bc$ when $a = 0$. For instance, when $b$ and $c$ are positive, the first edge points up and the second points right, a clockwise quarter turn, so the oriented area $-bc$ is negative. If $a = c = 0$, the first column is zero and both sides equal $0$.

Definition Let $A$ be a real $n \times n$ matrix with $n = 2$ or $n = 3$. For every bounded region $R$ in $\mathbb{R}^n$ that has an area (when $n = 2$) or a volume (when $n = 3$), the image $A(R) = \{A\mathbf{x} : \mathbf{x} \in R\}$ satisfies

where $\operatorname{vol}$ means area in the plane. The number $\lvert \det A \rvert$ is the volume scale factor of $A$, called the area scale factor in the plane. For $n \ge 4$ the same number is the scale factor of $n$-dimensional volume.

Definition Let $A = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 \end{bmatrix}$ be an invertible real $2 \times 2$ matrix. The map $\mathbf{x} \mapsto A\mathbf{x}$ is orientation-reversing if it sends the positively oriented pair $(\mathbf{e}_1, \mathbf{e}_2)$ to a negatively oriented pair $(\mathbf{a}_1, \mathbf{a}_2)$, that is, if $\det A < 0$. It is orientation-preserving if $\det A > 0$. An orientation-reversing map sends every positively oriented pair $(\mathbf{u}, \mathbf{v})$ to a negatively oriented pair $(A\mathbf{u}, A\mathbf{v})$, because

a case of the product rule proved in the next lesson. The same definitions apply to an invertible $3 \times 3$ matrix, with right-handed and left-handed triples. A singular matrix neither keeps nor reverses orientation, since its image vectors are dependent.

Definition A real $2 \times 2$ matrix $A$ has $\det A = 0$ if and only if its columns are linearly dependent. In that case $A$ sends the whole plane into a line through the origin, or onto the origin alone, and the image of the unit square has area $0$. This follows from the definition, since independent columns span a genuine parallelogram with positive area. Likewise, a real $3 \times 3$ matrix has determinant $0$ exactly when its columns are dependent, and then it sends space into a plane, a line or the single point $\mathbf{0}$, and the image of the unit cube has volume $0$.