Notation: For an
matrix
by
we mean the submatrix
of
which is obtained
by deleting the
row and
column.
EXAMPLE 2.6.1
Consider a matrix
Then
and
EXAMPLE 2.6.3
- Let
Then,
For example, for
- Let
Then,
For example, if
then
EXERCISE 2.6.4
- Find the determinant of the following matrices.
- Show that the determinant of a triangular matrix is the product
of its diagonal entries.
DEFINITION 2.6.5
A matrix
is said to be a singular matrix if
It is called non-singular if
The proof of the next theorem is omitted. The interested reader is
advised to go through Appendix 14.3.
THEOREM 2.6.6
Let
be an
matrix. Then
- if
is obtained from
by interchanging
two rows, then
,
- if
is obtained from
by
multiplying a row by
then
,
- if all the elements of one row or column of
are 0
then
,
- if
is obtained from
by replacing the
th
row by itself plus
times the
th row, where
then
,
- if
is a square matrix having two rows equal then
.
Remark 2.6.7
- Many authors define the determinant using
``Permutations." It turns out that THE WAY WE HAVE DEFINED DETERMINANT
is usually called the expansion of the determinant along the first row.
- Part 1 of
Lemma 2.6.6
implies that ``one can also calculate the determinant by expanding along
any row." Hence, for an
matrix
for every
,
one also has
Remark 2.6.8
- Let
and
be two vectors in
Then consider the parallelogram,
formed by the vertices
and
We
Recall that the dot product,
and
is the length of the vector
We
denote the length by
With the above notation, if
is the angle between the vectors
and
then
Which tells us,
Hence, the claim holds. That is, in
the determinant is
times
the area of the parallelogram.
- Let
and
be three elements of
Recall that the
cross product of two vectors in
is,
Note here that if
then
Let
be the parallelopiped formed with
as a vertex and
the vectors
as adjacent vertices. Then
observe that
is a vector perpendicular to the plane that
contains the parallelogram formed by the vectors
and
So, to
compute the volume of the parallelopiped
we need to look at
where
is the angle between the
vector
and the normal vector to the parallelogram formed
by
and
So,
Hence,
- Let
and let
be an
matrix.
Then the following properties of
also hold for the volume of
an
-dimensional parallelopiped formed with
as one vertex and the vectors
as adjacent vertices:
- If
and
then
Also, volume of a unit
-dimensional cube is
- If we replace the vector
by
for some
then
the determinant of the new matrix is
.
This is also true for the volume, as the
original volume gets multiplied by
- If
for some
then the vectors
will give rise to an
-dimensional
parallelopiped. So, this parallelopiped lies on an
-dimensional
hyperplane. Thus, its
-dimensional volume will be zero. Also,
In general, for any
matrix
it can be proved that
is indeed equal to the volume of the
-dimensional
parallelopiped. The actual proof is beyond the scope of this book.
Subsections
A K Lal
2007-09-12