### SectionDMDeterminant of a Matrix

Before we define the determinant of a matrix, we take a slight detour to introduce elementary matrices. These will bring us back to the beginning of the course and our old friend, row operations.

#### SubsectionEMElementary Matrices

Elementary matrices are very simple, as you might have suspected from their name. Their purpose is to effect row operations (Definition RO) on a matrix through matrix multiplication (Definition MM). Their definitions look much more complicated than they really are, so be sure to skip over them on your first reading and head right for the explanation that follows and the first example.

##### Definition ELEM Elementary Matrices

1. For $i\neq j$, $\elemswap{i}{j}$ is the square matrix of size $n$ with \begin{equation*} \matrixentry{\elemswap{i}{j}}{k\ell}= \begin{cases} 0 & k\neq i, k\neq j, \ell\neq k\\ 1 & k\neq i, k\neq j, \ell=k\\ 0 & k=i, \ell\neq j\\ 1 & k=i, \ell=j\\ 0 & k=j, \ell\neq i\\ 1 & k=j, \ell=i \end{cases} \end{equation*}
2. For $\alpha\neq 0$, $\elemmult{\alpha}{i}$ is the square matrix of size $n$ with \begin{equation*} \matrixentry{\elemmult{\alpha}{i}}{k\ell}= \begin{cases} 0 & \ell\neq k\\ 1 & k\neq i, \ell=k\\ \alpha & k=i, \ell=i \end{cases} \end{equation*}
3. For $i\neq j$, $\elemadd{\alpha}{i}{j}$ is the square matrix of size $n$ with \begin{equation*} \matrixentry{\elemadd{\alpha}{i}{j}}{k\ell}= \begin{cases} 0 & k\neq j, \ell\neq k\\ 1 & k\neq j, \ell=k\\ 0 & k=j, \ell\neq i, \ell\neq j\\ 1 & k=j, \ell=j\\ \alpha & k=j, \ell=i\\ \end{cases} \end{equation*}

Again, these matrices are not as complicated as their definitions suggest, since they are just small perturbations of the $n\times n$ identity matrix (Definition IM). $\elemswap{i}{j}$ is the identity matrix with rows (or columns) $i$ and $j$ trading places, $\elemmult{\alpha}{i}$ is the identity matrix where the diagonal entry in row $i$ and column $i$ has been replaced by $\alpha$, and $\elemadd{\alpha}{i}{j}$ is the identity matrix where the entry in row $j$ and column $i$ has been replaced by $\alpha$. (Yes, those subscripts look backwards in the description of $\elemadd{\alpha}{i}{j}$). Notice that our notation makes no reference to the size of the elementary matrix, since this will always be apparent from the context, or unimportant.

The raison d'etre for elementary matrices is to “do” row operations on matrices with matrix multiplication. So here is an example where we will both see some elementary matrices and see how they accomplish row operations when used with matrix multiplication.

##### Example EMROElementary matrices and row operations

The next three theorems establish that each elementary matrix effects a row operation via matrix multiplication.

##### Theorem EMDRO Elementary Matrices Do Row Operations

Suppose that $A$ is an $m\times n$ matrix, and $B$ is a matrix of the same size that is obtained from $A$ by a single row operation (Definition RO). Then there is an elementary matrix of size $m$ that will convert $A$ to $B$ via matrix multiplication on the left. More precisely,

1. If the row operation swaps rows $i$ and $j$, then $B=\elemswap{i}{j}A$.
2. If the row operation multiplies row $i$ by $\alpha$, then $B=\elemmult{\alpha}{i}A$.
3. If the row operation multiplies row $i$ by $\alpha$ and adds the result to row $j$, then $B=\elemadd{\alpha}{i}{j}A$.

Later in this section we will need two facts about elementary matrices.

##### Theorem EMN Elementary Matrices are Nonsingular

If $E$ is an elementary matrix, then $E$ is nonsingular.

Notice that we have now made use of the nonzero restriction on $\alpha$ in the definition of $\elemmult{\alpha}{i}$. One more key property of elementary matrices.

##### Theorem NMPEM Nonsingular Matrices are Products of Elementary Matrices

Suppose that $A$ is a nonsingular matrix. Then there exists elementary matrices $E_1,\,E_2,\,E_3,\,\dots,\,E_t$ so that $A=E_1 E_2 E_3\dots E_t$.

#### SubsectionDDDefinition of the Determinant

We will now turn to the definition of a determinant and do some sample computations. The definition of the determinant function is recursive, that is, the determinant of a large matrix is defined in terms of the determinant of smaller matrices. To this end, we will make a few definitions.

##### Definition SM SubMatrix

Suppose that $A$ is an $m\times n$ matrix. Then the submatrix $\submatrix{A}{i}{j}$ is the $(m-1)\times (n-1)$ matrix obtained from $A$ by removing row $i$ and column $j$.

##### Definition DM Determinant of a Matrix

Suppose $A$ is a square matrix. Then its determinant, $\detname{A}=\detbars{A}$, is an element of $\complex{\null}$ defined recursively by:

1. If $A$ is a $1\times 1$ matrix, then $\detname{A}=\matrixentry{A}{11}$.
2. If $A$ is a matrix of size $n$ with $n\geq 2$, then \begin{align*} \detname{A}&= \matrixentry{A}{11}\detname{\submatrix{A}{1}{1}} -\matrixentry{A}{12}\detname{\submatrix{A}{1}{2}} +\matrixentry{A}{13}\detname{\submatrix{A}{1}{3}}-\\ &\quad \matrixentry{A}{14}\detname{\submatrix{A}{1}{4}} +\cdots +(-1)^{n+1}\matrixentry{A}{1n}\detname{\submatrix{A}{1}{n}} \end{align*}

So to compute the determinant of a $5\times 5$ matrix we must build 5 submatrices, each of size $4$. To compute the determinants of each the $4\times 4$ matrices we need to create 4 submatrices each, these now of size $3$ and so on. To compute the determinant of a $10\times 10$ matrix would require computing the determinant of $10!=10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2=3,628,800$ $1\times 1$ matrices. Fortunately there are better ways. However this does suggest an excellent computer programming exercise to write a recursive procedure to compute a determinant.

Let us compute the determinant of a reasonably sized matrix by hand.

##### Example D33MDeterminant of a $3\times 3$ matrix

In practice it is a bit silly to decompose a $2\times 2$ matrix down into a couple of $1\times 1$ matrices and then compute the exceedingly easy determinant of these puny matrices. So here is a simple theorem.

##### Theorem DMST Determinant of Matrices of Size Two

Suppose that $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$. Then $\detname{A}=ad-bc$.

Do you recall seeing the expression $ad-bc$ before? (Hint: Theorem TTMI)

#### SubsectionCDComputing Determinants

There are a variety of ways to compute the determinant. We will establish first that we can choose to mimic our definition of the determinant, but by using matrix entries and submatrices based on a row other than the first one.

##### Theorem DER Determinant Expansion about Rows

Suppose that $A$ is a square matrix of size $n$. Then for $1\leq i\leq n$ \begin{align*} \detname{A}&= (-1)^{i+1}\matrixentry{A}{i1}\detname{\submatrix{A}{i}{1}}+ (-1)^{i+2}\matrixentry{A}{i2}\detname{\submatrix{A}{i}{2}}\\ &\quad+(-1)^{i+3}\matrixentry{A}{i3}\detname{\submatrix{A}{i}{3}}+ \cdots+ (-1)^{i+n}\matrixentry{A}{in}\detname{\submatrix{A}{i}{n}} \end{align*} which is known as expansion about row $i$.

We can also obtain a formula that computes a determinant by expansion about a column, but this will be simpler if we first prove a result about the interplay of determinants and transposes. Notice how the following proof makes use of the ability to compute a determinant by expanding about any row.

##### Theorem DT Determinant of the Transpose

Suppose that $A$ is a square matrix. Then $\detname{\transpose{A}}=\detname{A}$.

Now we can easily get the result that a determinant can be computed by expansion about any column as well.

##### Theorem DEC Determinant Expansion about Columns

Suppose that $A$ is a square matrix of size $n$. Then for $1\leq j\leq n$ \begin{align*} \detname{A}&= (-1)^{1+j}\matrixentry{A}{1j}\detname{\submatrix{A}{1}{j}}+ (-1)^{2+j}\matrixentry{A}{2j}\detname{\submatrix{A}{2}{j}}\\ &\quad+(-1)^{3+j}\matrixentry{A}{3j}\detname{\submatrix{A}{3}{j}}+ \cdots+ (-1)^{n+j}\matrixentry{A}{nj}\detname{\submatrix{A}{n}{j}} \end{align*} which is known as expansion about column $j$.

That the determinant of an $n\times n$ matrix can be computed in $2n$ different (albeit similar) ways is nothing short of remarkable. For the doubters among us, we will do an example, computing a $4\times 4$ matrix in two different ways.

##### Example TCSDTwo computations, same determinant

When a matrix has all zeros above (or below) the diagonal, exploiting the zeros by expanding about the proper row or column makes computing a determinant insanely easy.

##### Example DUTMDeterminant of an upper triangular matrix

When you consult other texts in your study of determinants, you may run into the terms “minor” and “cofactor,” especially in a discussion centered on expansion about rows and columns. We have chosen not to make these definitions formally since we have been able to get along without them. However, informally, a minor is a determinant of a submatrix, specifically $\detname{\submatrix{A}{i}{j}}$ and is usually referenced as the minor of $\matrixentry{A}{ij}$. A cofactor is a signed minor, specifically the cofactor of $\matrixentry{A}{ij}$ is $(-1)^{i+j}\detname{\submatrix{A}{i}{j}}$.