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anonymous
 5 years ago
a) If A + B is also invertible, then show that A^1 + B^1 is also invertible by finding a formula for it. Hint: Consider A^1(A+B)B^1 and use Theorem 1.39.
Theorem 1.39 If A and B are invertible nxn matrices, then AB is invertible and (AB)^−1 = (B^1)(A^1)
b) Generalize the previous result: If cA + dB is invertible, for real numbers c and d then show that dA−1 + cB−1 is also invertible by finding a formula for it. Cite any theorems or definitions used.
anonymous
 5 years ago
a) If A + B is also invertible, then show that A^1 + B^1 is also invertible by finding a formula for it. Hint: Consider A^1(A+B)B^1 and use Theorem 1.39. Theorem 1.39 If A and B are invertible nxn matrices, then AB is invertible and (AB)^−1 = (B^1)(A^1) b) Generalize the previous result: If cA + dB is invertible, for real numbers c and d then show that dA−1 + cB−1 is also invertible by finding a formula for it. Cite any theorems or definitions used.

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0**For a), I don't understand what they mean by finding a formula...and thanks :)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I'll give it a try. Just give me a minute.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0We are to assume that A and B are both nxn matrices?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0They are both nxn invertible matrices.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Well I think I got the answer of the part a.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0By the theorem you wrote above, we can see that: \[A^{1}(A+B)B^{1}\] is an invertible matrix, since it's multiplication of three invertible matrices.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Using properties of matrix multiplication, \[A^{1}(A+B)B^{1}=(A^{1}A+A^{1}B)B^{1}=A^{1}AB^{1}+A^{1}BB^{1}=B^{1}+A^{1}=A^{1}+B^{1}\] Clearly A^1+B^1 is equal to an invertible matrix, and hence it's also an invertible matrix.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Are you there meganchiu?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Does the answer make sense to you?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Would this have anything to do with it: Consider (A^1(A+B)B^1). (A^1(A+B)B^1)^1 = (B^1)^1(A+B)^1(A^1)^1 =B(A+B)^1(A) < ** Since B and A are invertible and since A+B is invertible, then ** is invertible Does that have anything to do with it?
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