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October 29th, 2009, 12:03 AM   #1
ced
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the inverse of a matrix

Im having trouble wrapping my head around this concept

Suppose A, B, X are nxn matrices with A, X, and A-AX invertible

(A - AX)^-1 = X^-1 * B

1. explain why B is invertible
2. solve for X

my answer
1. the product of nxn invertible matrices are invertible. (A-Ax) is an invertible matrix, which is the product of 2 invertible nxn matrices, X and B. Not the best logic but I have no idea what else to say
2. no idea where to start

thanks for the help
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October 29th, 2009, 04:19 AM   #2
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Re: the inverse of a matrix

Quote:
Originally Posted by ced
my answer
1. the product of nxn invertible matrices are invertible.
Correct.
Quote:
(A-Ax) is an invertible matrix, which is the product of 2 invertible nxn matrices, X and B. Not the best logic but I have no idea what else to say
Indeed not. It isn't necessarily true-- just because A is invertible doesn't mean that A=X*B => B is invertible
Try multiplying on the left by X. Then you can use the statement "the product of two invertible matrices is invertible".

Quote:
2. no idea where to start

thanks for the help
Pretend that A,B, and X are numbers instead of matrices, and solve that way, keeping the order straight. Then go back and rewrite "without division"-- by which I mean instead of dividing, multiply by the inverse. Of course, when you are rewriting it, you need to make sure all of your steps are justified.
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