Mathematical Methods for Physicists: A concise introduction - Site Map

THE INVERSE OF A MATRIX
or
0
1 0
10
A 2 B 3 A 3 B 2 0 B 2 B 2
B
C B
CB
@ A 3 B 1 A 1 B 3 A ˆ @ B 3 0 B 1 A@
A 1 B 2 A 2 B 1 B 2 B 1 0
A 1
A 2
A 3
1
C
A:
Thus the vector product may be represented as the product of a skew-symmetric
matrix and a column matrix. However, this de®nition only holds for 3 3
matrices.
Similarly, curl A may be represented in terms of a skew-symmetric matrix
operator, given in Cartesian coordinates by
0
10
1
rA ˆ
B
@
0 @=@x 3 @=@x 2
@=@x 3 0 @=@x 1
@=@x 2 @=@x 1 0
CB
A@
In a similar way, we can investigate the triple scalar product and the triple vector
product.
A 1
A 2
A 3
C
A:
The inverse of a matrix
If for a given square matrix ~ A there exists a matrix ~B such that ~ A ~B ˆ ~B ~ A ˆ ~I,
where ~I is a unit matrix, then ~B is called an inverse of matrix ~A.
Example 3.8
The matrix
is an inverse of
since
and
~B ˆ 3 5
1 2
2 5
~A ˆ
;
1 3
2 5 3 5
~A ~B ˆ
1 3 1 2
~B ~A ˆ 3 5
2 5
1 2 1 3
ˆ 1 0
ˆ ~I
0 1
ˆ 1 0
ˆ ~I:
0 1
An invertible matrix has a unique inverse. That is, if ~B and ~C are both inverses
of the matrix ~ A, then ~B ˆ ~C. The proof is simple. Since ~B is an inverse of ~ A,
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