From A First Course in Linear Algebra
Version 1.32
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/
First, a slight detour, as we introduce elementary matrices, which will bring us
back to the beginning of the course and our old friend, row operations.
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 more complicated than they really are, so be sure to read ahead after you read the definition for some explanations and an example.
Definition ELEM
Elementary Matrices
(This definition contains Notation ELEM.)
Again, these matrices are not as complicated as they appear, since they are mostly pertubations of the identity matrix (Definition IM). is the identity matrix with rows (or columns) and trading places, is the identity matrix where the diagonal entry in row and column has been replaced by , and is the identity matrix where the entry in row and column has been replaced by . (Yes, those subscripts look backwards in the description of ). 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’être 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 can accomplish row operations.
Example EMRO
Elementary matrices and row operations
We will perform a sequence of row operations (Definition RO) on the
matrix
,
while also multiplying the matrix on the left by the appropriate
elementary matrix.
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
is an
matrix, and
is a matrix of the same size that is obtained from
by a
single row operation (Definition RO). Then there is an elementary matrix of size
that will
convert
to via
matrix multiplication on the left. More precisely,
Proof In each of the three conclusions, performing the row operation on will create the matrix where only one or two rows will have changed. So we will establish the equality of the matrix entries row by row, first for the unchanged rows, then for the changed rows, showing in each case that the result of the matrix product is the same as the result of the row operation. Here we go.
Row of the product , where , , is unchanged from ,
Row of the product is row of ,
Row of the product is row of ,
So the matrix product is the same as the row operation that swaps rows and .
Row of the product , where , is unchanged from ,
Row of the product is times row of ,
So the matrix product is the same as the row operation that swaps multiplies row by .
Row of the product , where , is unchanged from ,
Row of the product , is times row of and then added to row of ,
So the matrix product is the same as the row operation that multiplies row by and adds the result to row .
Later in this section we will need two facts about elementary matrices.
Theorem EMN
Elementary Matrices are Nonsingular
If is an elementary
matrix, then is
nonsingular.
Proof We show that we can row-reduce each elementary matrix to the identity matrix. Given an elementary matrix of the form , perform the row operation that swaps row with row . Given an elementary matrix of the form , with , perform the row operation that multiplies row by . Given an elementary matrix of the form , with , perform the row operation that multiplies row by and adds it to row . In each case, the result of the single row operation is the identity matrix. So each elementary matrix is row-equivalent to the identity matrix, and by Theorem NMRRI is nonsingular.
Notice that we have now made use of the nonzero restriction on in the definition of . One more key property of elementary matrices.
Theorem NMPEM
Nonsingular Matrices are Products of Elementary Matrices
Suppose that
is a nonsingular matrix. Then there exists elementary matrices
so
that .
Proof Since is nonsingular, it is row-equivalent to the identity matrix by Theorem NMRRI, so there is a sequence of row operations that converts to . For each of these row operations, form the associated elementary matrix from Theorem EMDRO and denote these matrices by . Applying the first row operation to yields the matrix . The second row operation yields , and the third row operation creates . The result of the full sequence of row operations will yield , so
Other than the cosmetic matter of re-indexing these elementary matrices in the opposite order, this is the desired result.
We’ll 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 is
an matrix. Then
the submatrix
is the matrix
obtained from
by removing row
and column .
(This definition contains Notation SM.)
Example SS
Some submatrices
For the matrix
we have the submatrices
Definition DM
Determinant of a Matrix
Suppose is a square matrix.
Then its determinant, ,
is an element of
defined recursively by:
If is a
matrix,
then .
If is a
matrix of size
with ,
then
(This definition contains Notation DM.)
So to compute the determinant of a matrix we must build 5 submatrices, each of size . To compute the determinants of each the matrices we need to create 4 submatrices each, these now of size and so on. To compute the determinant of a matrix would require computing the determinant of 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’s compute the determinant of a reasonable sized matrix by hand.
Example D33M
Determinant of a
matrix
Suppose that we have the
matrix
Then
In practice it is a bit silly to decompose a matrix down into a couple of 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 .
Then
Proof Applying Definition DM,
Do you recall seeing the expression before? (Hint: Theorem TTMI)
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 is a
square matrix of size .
Then
which is known as expansion about row .
Proof First, the statement of the theorem coincides with Definition DM when , so throughout, we need only consider .
Given the recursive definition of the determinant, it should be no surprise that we will use induction for this proof (Technique I). When , there is nothing to prove since there is but one row. When , we just examine expansion about the second row,
So the theorem is true for matrices of size and . Now assume the result is true for all matrices of size as we derive an expression for expansion about row for a matrix of size . We will abuse our notation for a submatrix slightly, so will denote the matrix formed by removing rows and , along with removing columns and . Also, as we take a determinant of a submatrix, we will need to “jump up” the index of summation partway through as we “skip over” a missing column. To do this smoothly we will set
Now,
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 is a
square matrix. Then .
Proof
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 is a
square matrix of size .
Then
which is known as expansion about column .
Proof
That the determinant of an matrix can be computed in different (albeit similar) ways is nothing short of remarkable. For the doubters among us, we will do an example, computing a matrix in two different ways.
Example TCSD
Two computations, same determinant
Let
Then expanding about the fourth row (Theorem DER with ) yields,
while expanding about column 3 (Theorem DEC with ) gives
Notice how much easier the second computation was. By choosing to expand about the third column, we have two entries that are zero, so two determinants need not be computed at all!
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 DUTM
Determinant of an upper triangular matrix
Suppose that
We will compute the determinant of this matrix by consistently expanding about the first column for each submatrix that arises and does not have a zero entry multiplying it.
If 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’ve chosen not to make these definitions formally since we’ve been able to get along without them. However, informally, a minor is a determinant of a submatrix, specifically and is usually referenced as the minor of . A cofactor is a signed minor, specifically the cofactor of is .
C24 Doing the computations by hand, find the determinant of the matrix below.
Contributed by Robert Beezer Solution [1099]
C25 Doing the computations by hand, find the determinant of the matrix below.
Contributed by Robert Beezer Solution [1099]
C26 Doing the computations by hand, find the determinant of the matrix .
Contributed by Robert Beezer Solution [1100]
C24 Contributed by Robert Beezer Statement [1097]
We’ll expand about the first row since there are no zeros to exploit,
C25 Contributed by Robert Beezer Statement [1097]
We can expand about any row or column, so the zero entry in the middle of the
last row is attractive. Let’s expand about column 2. By Theorem DER and
Theorem DEC you will get the same result by expanding about a different row or
column. We will use Theorem DMST twice.
C26 Contributed by Robert Beezer Statement [1098]
With two zeros in column 2, we choose to expand about that column
(Theorem DEC),