Determinants

The determinant of a matrix $A$ , denoted $\det(A)$ , is only defined when $A$ is a square matrix, i.e., the number of rows is equal to the number of columns. If $A$ is an $n\times n$ matrix then $A$ may be regarded as a function of $F^n$ , sending vectors to points.

If $F=\mathbb{R}$ then it turns out the matrix $A$ , regarded as a function $A:\mathbb{R}^n\rightarrow \mathbb{R}^n$ , will send the unit hypercube $[0,1]\times ...\times [0,1]$ ($n$ times) in $\mathbb{R}^n$ to a parallelepiped (the $n$ -dimensional analog of a parallelogram). It is known that the absolute value of the determinant of $A$ measures the volume of that parallelpiped. For example, $$A=\left( \begin{array}{cc} 2&1\\ 1&2 \end{array} \right)$$ sends the vertices of the unit square $[0,1]\times [0,1]=\{(x,y) \vert 0\leq x\leq 1, 0\leq y\leq 1\}$ to the points $(0,0)$ , $(2,1)$ , $(1,2)$ , $(3,3)$ . These points bound a parallelogram having volume $\det(A)=3$ .

If $m=n=2$ the determinant is easy to define:

$$\det\left( \begin{array}{cc} a&b\\ c&d \end{array} \right)=ad-bc.$$

The easiest constructive way to define the determinant of an arbitrary $n\times n$ matrix $A=(a_{ij})$ is to use the Laplace cofactor expansion (along the $i^{th}$ row): for any $1\leq i\leq n$ , we have

$$\det(A)=\sum_{j=1}^n (-1)^{i+j}\det(A_{ij}),$$ (5.2)

where $A_{ij}$ is the $(n-1)\times (n-1)$ `submatrix of $A$ ' obtained by omitting all the entries in the $i^{th}$ row or the $j^{th}$ column. The matrix $A_{ij}$ is called the $(i,j)$ -minor of $A$ and $(-1)^{i+j}\det(A_{ij})$ is called the $(i,j)$ -cofactor.

The determinant has many remarkable properties. For example,

• You can factor an expression out of any row or column. For example,
  $$\det\left(\begin{array}{cc} ra & rb \ c & d\end{array} \right)= ... ...ay} \right) =r\det\left(\begin{array}{cc} a & b \ c & d\end{array} \right).$$

• The Lagrange expansion holds for any row or column. For example, the middle column expansion for a $3\times 3$ matrix:
  $$\begin{array}{c} \det\left(\begin{array}{ccc} a_{11} & a_{... ... a_{13} \\ a_{21} & a_{23}\end{array} \right). \end{array}$$

• The determinant, as a function of its rows or columns, is additive. For example,
  $$\det( \left(\begin{array}{cc} a+a' & b \ c+c' & d\end{array} \ri... ...right) + \det( \left(\begin{array}{cc} a' & b \ c'& d \end{array} \right).$$

For a proof, see any text on linear algebra (e.g., [JN]).

A square matrix $A$ is singular if $\det(A)=0$ . Otherwise, $A$ is called non-singular or invertible.

Example 5.5.1   Taking $i=2$ ,

$$\begin{array}{c} \det\left( \begin{array}{ccc} 1&2&3\ ... ...\right)\\ =(-4)(18-24)+(5)(9-21)+(-6)(8-14)=0. \end{array}$$

This implies $A$ is singular. Indeed, the parallelepiped generated by $(1, 2, 3)$ , $(4,5,6)$ , $(7,8,9)$ , must be flat ($2$ -dimensional, hence have 0 volume) since $(4,5,6)=(1,2,3)+(1,1,1)$ and $(7,8,9)=(1,2,3)+2(1,1,1)$ .

An important fact about singular matrices, and one that we will use later, is the following.

Lemma 5.5.2   Suppose $A$ is an $n\times n$ matrix with entries in $F$ . The following are equivalent:

• $\det(A)\not= 0$ ,

• there is no non-zero vector $v$ such that $Av=0$ , where 0 is the zero vector in $F^n$ ,

• $A^{-1}$ exists.

For a proof, see any text on linear algebra.

We end this section with one last key property of determinants.

Lemma 5.5.3   If $A,B$ are any two $n\times n$ matrices having entries in $F$ then $\det(AB)=\det(A)\det(B)$ .

More details on all this material can be found in any text book on linear algebra.