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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.

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  1. anonymous
    • 5 years ago
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    **For a), I don't understand what they mean by finding a formula...and thanks :)

  2. anonymous
    • 5 years ago
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    I'll give it a try. Just give me a minute.

  3. anonymous
    • 5 years ago
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    We are to assume that A and B are both nxn matrices?

  4. anonymous
    • 5 years ago
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    Yup!

  5. anonymous
    • 5 years ago
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    They are both nxn invertible matrices.

  6. anonymous
    • 5 years ago
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    Well I think I got the answer of the part a.

  7. anonymous
    • 5 years ago
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    By 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.

  8. anonymous
    • 5 years ago
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    Using 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.

  9. anonymous
    • 5 years ago
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    Are you there meganchiu?

  10. anonymous
    • 5 years ago
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    yup im here

  11. anonymous
    • 5 years ago
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    Does the answer make sense to you?

  12. anonymous
    • 5 years ago
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    Would 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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