Orange Coast College: MATH 285 ROBERT

. 2
3 : Properties of determinants MATH 285 -
9/10/2025


Result :
if A = (aij]nxn M
is lower or uper triangular
DetCA)
=
then
ii =
11922933 ... and




Pf Suppose A is U T



↑aea S
: .

-




UT -
aij = 0 when is j



DET(A) : EJCP, ...
Pn) a , p, 9292939s --. an Don


=
911922933 ... Ann+ STUFF

All non privial (p , ... Pn) SO STUFF = O

Produce O for that term




3x3
S Gla)
-O
IA) = DETCA) =
911922933 + T( , 3, 2) all
as
+ T(213)
aas ...
Yo

PROPERTIES :


1
. If B =
PijA , DETCB) = -DETCA) *

. If
2 B = MICK) A
,
DETCB) = 1 . DETCA) a


3 .
If B =
AijCkA ,
DET(B) = DETCA) **




DET0)
( ()
g
E :



DET O
+ =
-
I




·
& : DET (PijA) = - DETCA)

·
P2 : DET (MiCK)A) = IDETCA)
& .
P3 : DET (Aij(k) A) = DETCA)

·
PP : DET(KA) = KVDETCA)
·5 :
DETCAY) DETCA)
①
=



This study
· (Strange
16 :source was downloaded by 100000900706475 from CourseHero.com on 10-07-2025 23:33:30 GMT -05:00


https://www.coursehero.com/file/251495799/MATH-285-ROBERT-09-10-2025pdf/

, Let at a , ..., rows of A


If ai b + = then with
>
-




/c
= then DETCA)= DETCB) + DET(C)




· P7 :
If A has a now of all O's

DET(A) : 0 ( if A has A COL of all O's (
· Pr :
if 2 now cor cols) of A are equal , (or mult's each other (
DET (A) = 0

· Pg : DET(AB) = DETCA) DET(B)

· Pir :
if A is
non-sing , DETCA)
= A

1 = DET (In) : DETCA A "( = DETCA") . DET(A)

So DETCA"): 1
DETCA)



Proofs :
-




· P : B : PrsA

B :
(bij] bij = aij i trs


&
,



asji = r



arj i = S

DET (B) = E+ (p, ..
.,
Pr ... Ps ...
Pn) bip ...
brp... bsps--- bupn
,


· o .
= - DET (A)

3 .
2 : 1-21

37 44 -
: DETCA) : 5 DETCB) = 3
DET [C2B) "CAB)T



DEF : We say Anxn is orthogonal if
-
A = AT
>
- If

Problem :
if A " = AT show DETCA) = =
A AT.
= In

CAT)
This study source1 was
: DET CA AT)
downloaded = DET(A) DET
by 100000900706475 from CourseHero.com on 10-07-2025 23:33:30 GMT -05:00
.


②
https://www.coursehero.com/file/251495799/MATH-285-ROBERT-09-10-2025pdf/