Question

If A,B,C,D are all invertible nxn matrices,solve for B
AB^-1=D?

I understand to solve for C you need to multiply each side with the inverse, but I dont understand the premultiply and postmultiply rules

Answer 1

Sorry i typed that wrong its
AB^-1C=D solve for B

Answer 2

Hello! :)

in AB^-1 C = D,
you need to isolate B, right??

Also, do you understand that in Matrix algebra,
AB and BA are different and not equal everytime?

Answer 3

lets take a simpler problem

XY = Z

isolate Y

you would pre multiply by X or post multiply by X?

Answer 4

you would multiply each side with X^-1 correct?

Answer 5

pre multiply

Answer 6

yes, i meant X^-1 sorry,

pre multiply or post multiply?

Answer 7

ok good

Answer 8

if you had post multiplied it,
XYX^-1 = Z X^-1

you could not have combined X and X^-1 to get I (the Identity matrix)

Answer 9

now lets go to our problem,

AB^-1 C = D

pre multiply by C or post? :)

Answer 10

damn, i mean C^-1 :P

Answer 11

haha post right?

Answer 12

yes!
do it, and post some steps?

Answer 13

AB^-1CC^-1=DC^-1
A^-1AB^-1=A^-1DC^-1

I get to
B^-1=A^-1DC^-1
but im not sure what i should do from here

Answer 14

very good! you solved more than half of it!

now take inverse on both sides...

Answer 15

and use this recursively,
\(\large (PQ)^{-1} = Q^{-1}P^{-1}\)

Answer 16

perfect so the answer would be
B=D^-1CA ?

Answer 17

how did u get that ?

Answer 18

\([A^{-1}D C^{-1}]^{-1} = (C^{-1})^{-1} D^{-1} (A^{-1})^{-1} = CD^{-1}A \)

Answer 19

oh sorry my bad I was looking at the wrong work I was doing before

this is my first time using this site and it's very helpful thank you for your help!!

Answer 20

hey, most welcome ^_^
happy to help :)

feel free to message me if you wanna ask anything about this site :)

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