082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

276 Chapter 8 Matrix algebra
( )
a b
If A is the matrix , we write its determinant as
c d
∣ a b
c d∣ . Note that the straight
lines || indicate that we are discussing the determinant, which is a scalar, rather than
the matrix itself. If the matrixAis such that |A| = 0, then it has no inverse and is said
tobe singular. If |A| ≠ 0 thenA −1 exists andAissaidtobe non-singular.
Asingular matrixAhas |A| = 0.
Anon-singular matrixAhas |A| ≠ 0.
Example8.23 IfA =
( ) 1 2
andB=
5 0
( ) −1 2
find |A|, |B|and |AB|.
−3 1
Solution |A| =
∣ 1 2
5 0∣ = (1)(0) − (2)(5) = −10
|B| =
∣ −1 2
−3 1∣ = (−1)(1) − (2)(−3) =5
( )( ) ( )
1 2 −1 2 −7 4
AB =
=
5 0 −3 1 −5 10
|AB| = (−7)(10) − (4)(−5) = −50
We note that |A||B| = |AB|.
The resultobtained inExample 8.23 istruemore generally:
IfAandBaresquare matrices of the same order, |A||B| = |AB|.
8.6.2 Orthogonalmatrices
A non-singular square matrixAsuch thatA T = A −1 is said to be orthogonal. Consequently,
ifAisorthogonalAA T =A T A =I.
Example8.24 Find the inverse ofA =
( ) 0 −1
. Deduce thatAisan orthogonal matrix.
1 0
Solution From the formula forthe inverse of a 2 ×2 matrix wefind
A −1 = 1 ( ) ( ) 0 1 0 1
=
1 −1 0 −1 0
This isclearly equal tothe transpose ofA. HenceAisan orthogonal matrix.
To findthe inversesoflargermatricesweshallneedtostudydeterminantsfurther.This
isdone inSection8.7.