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1. A3(A3+A2)-1+A2(A2+A)-1+2(A3+A2)-1A4

 

2. -2 A-2(A-2+A-3)-1+2A3(A2+A3)-1+ 3/4 A(A-1+I)-1+ 3/4 A3(A+I)-1

 Aug 20, 2017
edited by syncstar  Aug 20, 2017
 #1
avatar+26396 
+2

1. A3(A3+A2)-1+A2(A2+A)-1+2(A3+A2)-1A4

Question see also:

https://web2.0calc.com/questions/matrix-problem-simplify-as-little-as-possible-each


1.
A3(A3+A2)1=A3[AA2+IA2]1|Identity matrix I =A3[(A+I)A2]1|Formula:(AB)1=B1A1=A3[(A2)1(A+I)1]|(A2)1=(AA)1=A1A1=(A1)2=A3(A1)2(A+I)1=AA2(A1)2(A+I)1|A2(A1)2=I=AI(A+I)1|AI=A=A(A+I)1

 

2.
A2(A2+A)1=A2(AA+IA)1|Identity matrix I =A2[(A+I)A]1|Formula:(AB)1=B1A1=A2[A1(A+I)1]=AAA1(A+I)1|AA1=I=AI(A+I)1|AI=A=A(A+I)1

 

3.
2(A3+A2)1A4=2[A2A+A2I]1A4|Identity matrix I =2[A2(A+I)]1A4|Formula:(AB)1=B1A1=2[(A+I)1(A2)1]A4|(A2)1=(AA)1=A1A1=(A1)2=2(A+I)1(A1)2A4=2(A+I)1(A1)2A2A2|(A1)2A2=I=2(A+I)1IA2|IA2=A2=2(A+I)1A2

 

summary
A3(A3+A2)1+A2(A2+A)1+2(A3+A2)1A4=A(A+I)1+A(A+I)1+2(A+I)1A2=2A(A+I)1+2(A+I)1A2=2A(A+I)1I+2(A+I)1A2|I=A1A=2A(A+I)1A1A+2(A+I)1A2=[A(A+I)1A1+(A+I)1A] 2A|A(A+I)1=[(A+I)A1]1={[(A+I)A1]1A1+(A+I)1A} 2A|(A+I)1A=[A1(A+I)]1={[(A+I)A1]1A1+[A1(A+I)]1} 2A=[(AA1+IA1)1A1+(A1A+A1I)1] 2A|AA1=A1A=IIA1=A1I=A1=[(I+A1)1A1+(I+A1)1] 2A=[(I+A1)1(A1+I)] 2A=[(I+A1)1(I+A1)] 2A|(I+A1)1(I+A1)=I=I2A=2A

 

 

laugh

 Aug 21, 2017
edited by heureka  Aug 21, 2017
 #2
avatar+26396 
+2

2. -2 A-2(A-2+A-3)-1+2A3(A2+A3)-1+ 3/4 A(A-1+I)-1+ 3/4 A3(A+I)-1

 

2A2(A2+A3)1+2A3(A2+A3)1+34A(A1+I)1+34A3(A+I)1= ?

 

1.
2A2(A2+A3)1+2A3(A2+A3)1(A2+A3)1=[(A1+I)A2]1(A2+A3)1=[(A+I)A2]1=2A2[(A1+I)A2]1+2A3[(A+I)A2]1|A2=A1A1=(AA)1=(A2)1=2A2[(A1+I)(A2)1]1+2A3[(A+I)A2]1|[(A1+I)(A2)1]1=[A2(A1+I)1]=2A2[A2(A1+I)1]+2A3[(A+I)A2]1=2A2A2(A1+I)1+2A3[(A+I)A2]1|A2A2=A1A1AA=A1(A1A)A=A1(I)A=A1A=I=2I(A1+I)1+2A3[(A+I)A2]1=2(A1+I)1+2A3[(A+I)A2]1|[(A+I)A2]1=[(A2)1(A+I)1]=2(A1+I)1+2A3[(A2)1(A+I)1]|(A2)1=A2=2(A1+I)1+2A3A2(A+I)1=2(A1+I)1+2AA2A2(A+I)1|A2A2=I=2(A1+I)1+2AI(A+I)1=2(A1+I)1+2A(A+I)1|A(A+I)1=[(A+I)A1]1=2(A1+I)1+2[(A+I)A1]1=2(A1+I)1+2(AA1+IA1)1=2(A1+I)1+2(I+A1)1=2(A1+I)1+2(A1+I)1=02A2(A2+A3)1+2A3(A2+A3)1=0

 

2.
2A2(A2+A3)1+2A3(A2+A3)1=2(A1+I)1+2A(A+I)1=02(A1+I)1+2A(A+I)1=02(A1+I)1=2A(A+I)1(A1+I)1=A(A+I)1

 

3.

2A2(A2+A3)1+2A3(A2+A3)1+34A(A1+I)1+34A3(A+I)1=0+34A(A1+I)1+34A3(A+I)1=34A(A1+I)1=A(A+I)1+34A3(A+I)1=34AA(A+I)1+34A3(A+I)1=34A2(A+I)1+34A3(A+I)1=34(A2+A3)(A+I)1=34A2(I+A)(A+I)1=34A2(I+A)(I+A)1=I=34A2I=34A2

 

laugh

 Aug 21, 2017

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