Determinant 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.

1. Construct the elementary matrix that will effect the row operation $\rowopadd{-6}{2}{3}$ on a $4\times 7$ matrix.
2. \begin{bmatrix} 2&3&-1\\ 3&8&2\\ 4&-1&-3 \end{bmatrix}
3. \begin{bmatrix} 3 & 9 & -2 & 4 & 2 \\ 0 & 1 & 4 & -2 & 7 \\ 0 & 0 & -2 & 5 & 2 \\ 0 & 0 & 0 & -1 & 6 \\ 0 & 0 & 0 & 0 & 4 \end{bmatrix}

Elementary matrices are very simple, as you might have suspected from their name. Their purpose is to effect row operations () on a matrix through matrix multiplication (). 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.

Again, these matrices are not as complicated as their definitions suggest, since they are just small perturbations of the $n\times n$ identity matrix (). $\elemswap{i}{j}$ 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trading places, $\elemmult{\alpha}{i}$ is the identity matrix where the diagonal entry in row $i$ and column $i$ has been replaced by $\alpha$, and $\elemadd{\alpha}{i}{j}$ is the identity matrix where the entry in row $j$ and column $i$ has been replaced by $\alpha$.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.

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

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

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.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

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.

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.

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.

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

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

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.

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.

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

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.

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.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.