Associativity. This rule holds for all matrices, and
hence for invertible matrices in particular.
Inverses. The inverse of a matrix A is itself invertible,
and so belongs to the set: (A
–1 )
–1 = A.
A study of DETERMINANTs and CRAMER’S RULE
shows that a matrix is invertible if, and only if, its
determinant is not zero. Thus the nth-order general linear group may be defined as the set of all n × n square
matrices with nonzero determinants. The subset of all
square matrices with determinant equal to 1 forms a
subgroup of GL n called the special linear group,
denoted SL n .
geodesic The shortest curve connecting two points
on a surface, and lying wholly on that surface, is called
a geodesic. For example, PYTHAGORAS’S THEOREM
shows that a straight line gives the shortest path
between two points on a plane, and thus straight lines
are the geodesics of a plane. On a SPHERE, geodesics are
sections of great circles. (Planes fly along geodesic arcs
across the Earth’s surface.) One envisions a geodesic as
the path a band of stretched elastic would adopt if held
at the two points in question and forced to remain on
the surface under study.
On a plane there is always only one geodesic
between any two given points. On a sphere there are
infinitely many geodesics that connect the two poles of
the sphere, for example. (If the two points chosen are
not antipodal, however, then the geodesic connecting
them is unique.)
A geodesic dome is a domelike structure made of
straight-line structural elements held in tension. The
straight line segments approximate geodesics of the
dome.
geometric distribution See BINOMIAL DISTRIBUTION.
geometric mean See MEAN.
geometric sequence (geometric progression) A SEQUENCE of numbers in which each term, except the
first, is a fixed multiple of the previous one is called a
geometric sequence. The constant ratio of terms is
called the common ratio. For example, the sequence
1,3,9,27,… is geometric, with common ratio 3. The
sequence 1, , , ,... is also geometric, with common
ratio .
A geometric sequence with first term a and common ratio r has the form: a, ar, ar
2
, ar
3
,… The nth term
a n of the sequence is given by a n = ar
n–1
. (It is common,
however, to start the count by calling the first term of
the sequence the “zero-th term,” so that a 0 = ar
0 = a
and the nth term of the sequence is given by the formula: a n = ar
n
.) A geometric sequence can also be
described as exponential. The common ratio r is the
base of the EXPONENTIAL FUNCTION f(n) = ar
n .
If the value r lies between –1 and 1, then the terms
ar
n of geometric sequence approach the value 0 as n
becomes large. If r = 1, then the geometric sequence is
the constant sequence a,a,a,… For all other values of r,
the sequence diverges.
The sum of the terms of a geometric sequence is
called a geometric series:
a + ar + ar
2 + ar
3 +…
In the study of CONVERGENT SERIES, the ratio test shows
that this series sums to a finite value if –1 < r < 1. The
value S of the sum can be computed as follows:
If S = a + ar + ar
2 + ar
3 +…, then rS = ar + ar
2 + ar
3 + ar
4
+… Subtracting yields: S – rS = a, and so S =
In particular we have 1 + r + r
2 + r
3 + … =
.
If r is a fraction of the form
(with N ≥ 2), then this
formula can be rewritten:
This particular expression can also be justified with a
physical demonstration, which we illustrate here for
the case N = 3:
John takes a piece of paper and tears it into
thirds. He hands one piece to Andrea, another
1
1
1
1
1
1
1
1
1
2
3
N N
N
N
N
N
+
+
+ =
−
= −
L
1
––
N
1
––
1 – r
1
––
1 – r
1
––
2
1
––
8
1
––
4
1
––
2
geometric sequence 223
hence for invertible matrices in particular.
Inverses. The inverse of a matrix A is itself invertible,
and so belongs to the set: (A
–1 )
–1 = A.
A study of DETERMINANTs and CRAMER’S RULE
shows that a matrix is invertible if, and only if, its
determinant is not zero. Thus the nth-order general linear group may be defined as the set of all n × n square
matrices with nonzero determinants. The subset of all
square matrices with determinant equal to 1 forms a
subgroup of GL n called the special linear group,
denoted SL n .
geodesic The shortest curve connecting two points
on a surface, and lying wholly on that surface, is called
a geodesic. For example, PYTHAGORAS’S THEOREM
shows that a straight line gives the shortest path
between two points on a plane, and thus straight lines
are the geodesics of a plane. On a SPHERE, geodesics are
sections of great circles. (Planes fly along geodesic arcs
across the Earth’s surface.) One envisions a geodesic as
the path a band of stretched elastic would adopt if held
at the two points in question and forced to remain on
the surface under study.
On a plane there is always only one geodesic
between any two given points. On a sphere there are
infinitely many geodesics that connect the two poles of
the sphere, for example. (If the two points chosen are
not antipodal, however, then the geodesic connecting
them is unique.)
A geodesic dome is a domelike structure made of
straight-line structural elements held in tension. The
straight line segments approximate geodesics of the
dome.
geometric distribution See BINOMIAL DISTRIBUTION.
geometric mean See MEAN.
geometric sequence (geometric progression) A SEQUENCE of numbers in which each term, except the
first, is a fixed multiple of the previous one is called a
geometric sequence. The constant ratio of terms is
called the common ratio. For example, the sequence
1,3,9,27,… is geometric, with common ratio 3. The
sequence 1, , , ,... is also geometric, with common
ratio .
A geometric sequence with first term a and common ratio r has the form: a, ar, ar
2
, ar
3
,… The nth term
a n of the sequence is given by a n = ar
n–1
. (It is common,
however, to start the count by calling the first term of
the sequence the “zero-th term,” so that a 0 = ar
0 = a
and the nth term of the sequence is given by the formula: a n = ar
n
.) A geometric sequence can also be
described as exponential. The common ratio r is the
base of the EXPONENTIAL FUNCTION f(n) = ar
n .
If the value r lies between –1 and 1, then the terms
ar
n of geometric sequence approach the value 0 as n
becomes large. If r = 1, then the geometric sequence is
the constant sequence a,a,a,… For all other values of r,
the sequence diverges.
The sum of the terms of a geometric sequence is
called a geometric series:
a + ar + ar
2 + ar
3 +…
In the study of CONVERGENT SERIES, the ratio test shows
that this series sums to a finite value if –1 < r < 1. The
value S of the sum can be computed as follows:
If S = a + ar + ar
2 + ar
3 +…, then rS = ar + ar
2 + ar
3 + ar
4
+… Subtracting yields: S – rS = a, and so S =
In particular we have 1 + r + r
2 + r
3 + … =
.
If r is a fraction of the form
(with N ≥ 2), then this
formula can be rewritten:
This particular expression can also be justified with a
physical demonstration, which we illustrate here for
the case N = 3:
John takes a piece of paper and tears it into
thirds. He hands one piece to Andrea, another
1
1
1
1
1
1
1
1
1
2
3
N N
N
N
N
N
+
+
+ =
−
= −
L
1
––
N
1
––
1 – r
1
––
1 – r
1
––
2
1
––
8
1
––
4
1
––
2
geometric sequence 223