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.
**For a), I don't understand what they mean by finding a formula...and thanks :)
I'll give it a try. Just give me a minute.
We are to assume that A and B are both nxn matrices?
Yup!
They are both nxn invertible matrices.
Well I think I got the answer of the part a.
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.
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.
Are you there meganchiu?
yup im here
Does the answer make sense to you?
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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