Matakuliah
Tahun
: D0024/Matematika Industri II
: 2008
Matematika
Pertemuan 23
Invers dan Determinan
The inverse of a square matrix
, sometimes called a
reciprocal matrix, is a matrix
such that
where is the identity matrix. Courant and Hilbert
(1989, p. 10) use the notation
to denote the inverse
matrix.
A square matrix
has an inverse iff the
determinant
(Lipschutz 1991, p. 45). A matrix
possessing an inverse is called nonsingular, or
invertible.
Bina Nusantara
(
1
)
For a
matrix
the matrix inverse is
(
3
)
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(
2
)
For a
matrix
(
5
)
the matrix inverse is
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A general
matrix can be inverted using methods such as the
Gauss-Jordan elimination, Gaussian elimination, or LU
decomposition.
The inverse of a product
of matrices
and
can be
expressed in terms of
and
. Let
Then
(
8
)
and
Bina Nusantara
(
7
)
Therefore,
so
(
where
1
1
)
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is the identity matrix, and
Determinant Theorem
Given a square matrix
, the following are equivalent:
1.
.
2. The columns of
are linearly independent.
3. The rows of
are linearly independent.
4. Range(
)=
.
5. Null(
)=
.
6.
has a matrix inverse.
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Determinant
A
determinant is defined to be
(
6
)
A
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determinant can be expanded "by minors" to obtain
A general determinant for a matrix
with no implied summation over
the cofactor of
defined by
(
9
)
has a value
and where
(also denoted
) is
and
is the minor of matrix
formed by eliminating
row and column
from
. This process is called determinant
expansion by minors (or "Laplacian expansion by minors,"
sometimes further shortened to simply "Laplacian expansion").
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For example, with
, the permutations and the
number of inversions they contain are 123 (0), 132 (1),
213 (1), 231 (2), 312 (2), and 321 (3), so the determinant
is given by
Bina Nusantara
• Kerjakan latihan dalam modul soal
Bina Nusantara