Understanding Engineering Mathematics

(やまだぃちぅ) #1
Problem 13.12
Find the determinant and adjoint of

A=

[− 123
011
− 102

]

and evaluateAAdjA

|A|=




∣∣

− 123
011
− 102




∣∣=−^1 (^2 )−^2 (^1 )+^3 (^1 )

=− 1

AdjA=

      

∣∣

11
02


∣∣
∣ −


∣∣

01
− 12


∣∣


∣∣

01
− 10


∣∣






23
02









− 13
− 12




∣ −





− 12
− 10

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

23
11




∣ −





− 13
01









− 12
01





      
T

Note, for example, that the element−






− 12
− 10




∣in the 2, 3 position of the matrix about to

be transposed is indeed the cofactor of the elementa 23 inAand check the other elements
similarly.


=

[ 2 − 11
− 41 − 2
− 11 − 1

]T

=

[ 2 − 4 − 1
− 111
1 − 2 − 1

]

We t h e n fi n d


AAdjA=

[− 123
011
− 102

][ 2 − 4 − 1
− 111
1 − 2 − 1

]

=

[− 100
0 − 10
00 − 1

]

=− 1

[ 100
010
001

]

=− 1 I
=|A|I

This is an example of the general result


A

AdjA
|A|

=I

which we will return to below.

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