Determinants and Their Geometric Meaning

Definition The determinant of a $2 \times 2$ matrix is

Determinants are defined only for square matrices. Vertical bars around the entries mean the determinant, not an absolute value.

Definition Let $A$ be a $2 \times 2$ matrix with columns $\mathbf{u}$ and $\mathbf{v}$. Draw $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{u} + \mathbf{v}$ from the origin. The parallelogram with corners at the origin and at the tips of these three vectors has area $\lvert \det A \rvert$. The map $\mathbf{x} \mapsto A\mathbf{x}$ sends the unit square, with corners $(0, 0)$, $(1, 0)$, $(1, 1)$ and $(0, 1)$, onto this parallelogram, and it multiplies the area of every region by the same factor $\lvert \det A \rvert$. The number $\det A$ itself, with its sign, is called the signed area of the parallelogram spanned by $\mathbf{u}$ and $\mathbf{v}$ in that order. Swapping the two columns changes its sign but not the area.

Definition Let $\mathbf{u}$ and $\mathbf{v}$ be the columns of a $2 \times 2$ matrix $A$. If $\det A > 0$, turning $\mathbf{u}$ counterclockwise through an angle strictly between $0$ and $\pi$ brings it to the direction of $\mathbf{v}$, just as $\mathbf{i}$ turns to $\mathbf{j}$, and the map $\mathbf{x} \mapsto A\mathbf{x}$ preserves orientation. If $\det A < 0$, the shorter turn from $\mathbf{u}$ to $\mathbf{v}$ is clockwise and the map reverses orientation. If $\det A = 0$, the vectors $\mathbf{u}$ and $\mathbf{v}$ lie on one line through the origin.

Definition The determinant of a $3 \times 3$ matrix $A = [a_{ij}]$ is

Each small determinant comes from deleting the row and the column of the entry in front of it, and the signs alternate $+$, $-$, $+$.

Definition Let $A = [a_{ij}]$ be an $n \times n$ matrix with $n \ge 2$. The minor $M_{ij}$ is the determinant of the $(n - 1) \times (n - 1)$ matrix left after deleting row $i$ and column $j$ of $A$, where a $1 \times 1$ matrix $[a]$ has determinant $a$. The cofactor is $C_{ij} = (-1)^{i + j} M_{ij}$. For any fixed row $i$ and any fixed column $j$

so every row and every column gives the same value. For $n = 2$ and $n = 3$ this agrees with the formulas above. For $n \ge 4$ the expansion along the first row is taken as the definition of $\det A$, since its minors are determinants of smaller matrices that are already defined. For a $3 \times 3$ matrix the signs $(-1)^{i + j}$ form a checkerboard,

Definition Let $A$ be a $3 \times 3$ matrix with columns $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$. The parallelepiped with edges $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$ from the origin has volume $\lvert \det A \rvert$, and the map $\mathbf{x} \mapsto A\mathbf{x}$ multiplies the volume of every solid by $\lvert \det A \rvert$. The sign gives the orientation. The determinant is positive when $\mathbf{u}$, $\mathbf{v}$, $\mathbf{w}$ form a right-handed triple like the coordinate directions, negative when they form a left-handed triple, the mirror image of a right-handed one, and $0$ when the three vectors lie in one plane through the origin. For square matrices $A$ and $B$ of the same size

matching the fact that doing $B$ and then $A$ scales first by $\det B$ and then by $\det A$.

Definition For a $2 \times 2$ or $3 \times 3$ matrix $A$, $\det A = 0$ exactly when the columns are linearly dependent, which means that some column is a sum of multiples of the other columns. For $2 \times 2$ this means one column is a multiple of the other, a zero column included. For $3 \times 3$ it means the three columns lie in one plane through the origin. Equivalently, the map $\mathbf{x} \mapsto A\mathbf{x}$ flattens the plane onto a line or a point, or space onto a plane, a line or a point, and there is a nonzero vector $\mathbf{x}$ with $A\mathbf{x} = \mathbf{0}$. The same holds with rows in place of columns, so a matrix with a zero row or two equal rows has determinant $0$.

Definition A square matrix $A$ is singular when $\det A = 0$ and nonsingular when $\det A \ne 0$. For an $n \times n$ matrix $A$ the following hold.

1. If $A$ is nonsingular, the system $A\mathbf{x} = \mathbf{b}$ has exactly one solution for every $\mathbf{b}$ in $\mathbb{R}^n$.
2. If $A$ is singular, then for each $\mathbf{b}$ the system $A\mathbf{x} = \mathbf{b}$ has either no solution or infinitely many, and $A\mathbf{x} = \mathbf{0}$ has infinitely many.

The next lesson shows that the nonsingular matrices are exactly the ones that have an inverse, which is why many books define a singular matrix as one with no inverse.

Definition Let $A$ be an $n \times n$ matrix with $\det A \ne 0$, and let $A_i$ be the matrix obtained from $A$ by replacing column $i$ with $\mathbf{b}$. Then the unique solution of $A\mathbf{x} = \mathbf{b}$ is

The rule says nothing when $\det A = 0$. It is an optional method, handy for a $2 \times 2$ system or for finding a single unknown, but for larger systems elimination takes far less arithmetic.