ask questions - get instant answers - receive/send p2p payments - get tips - get points - get fundraising support - meet community tutors
First time here? Checkout the FAQs!
x
0 like 0 dislike
570 views
Let \(A\) and \(B\) be \(n \times n\) matrices. If \(A+B\) is invertible, show that \(A(A+B)^{-1} B=\) \(B(A+B)^{-1} A\). [Don't assume that \(A B=B A\) ].
in Mathematics by Platinum (108k points) | 570 views

2 Answers

0 like 0 dislike

A(A+B)-1B = (A+B - B)(A+B)-1B = (A+B)(A+B)-1B - B(A+B)-1B = B - B(A+B)-1B ;

B(A+B)-1A = B(A+B)-1(A+B - B) = B(A+B)-1(A+B) - B(A+B)-1B = B - B(A+B)-1B.

by (104 points)
0 like 0 dislike
To show that \(A(A+B)^{-1} B=B(A+B)^{-1} A\), we will use the following property of the inverse of a matrix: Given an invertible matrix \(X\), if there exists a matrix \(Y\) such that \(X Y=\) \(I\), then \(Y=X^{-1}\).

Let's denote the matrix \((A+B)^{-1}\) as \(C\). Then, we can rewrite the expression we need to prove as: \(A C B=B C A\).
Now let's compute the product of the matrices \(A C B\) and \(B C A\) :
\((A C B)(B C A)=A(C B)(C A)=A((C B) C) A\) (associativity of matrix multiplication)
Since matrix multiplication is associative, we can rewrite the expression inside the parentheses as:
\[
A((C B) C) A=A(C(B C)) A
\]
Now, notice that we can use the property of the inverse we mentioned earlier:
Since \(A+B=A C+B C\) and \(C\) is the inverse of \(A+B\), we have:

\[
(A+B) C=(A C+B C) C=I
\]
Thus,
\[
B C=(A+B) C-A C=I-A C
\]
Now, we can substitute this back into our expression:
\[
A(C(B C)) A=A(C(I-A C)) A
\]
Using the distributive property of matrix multiplication, we have:
\[
A(C I-C A C) A=A(C-C A C) A
\]
Since \(C I=C\), we get:
\[
A(C-C A C) A=A(C-C A C) A
\]
This shows that:

\[
A C B=B C A
\]
Therefore, we have proven that \(A(A+B)^{-1} B=B(A+B)^{-1} A\).
by Platinum (108k points)

Related questions

0 like 0 dislike
0 answers
0 like 0 dislike
0 answers

Join MathsGee for AI-powered Q&A, tutor insights, P2P payments, interactive education, live lessons, and a rewarding community experience.

On MathsGee, you can:

1. Ask Questions on Various Topics


2. Request a Tutor


3. Start a Fundraiser


4. Become a Tutor


5. Create Tutor Session - For Verified Tutors


6. Host Tutor Session - For Verified Tutors


7. Join Tutor Session


8. Enjoy our interactive learning resources


9. Get tutor-verified answers

10. Vote on questions and answers

11. Tip/Donate your favorite community members

12. Earn points by participating


Posting on the MathsGee

1. Remember the human

2. Act like you would in real life

3. Find original source of content

4. Check for duplicates before publishing

5. Read the community guidelines


MathsGee Rules

1. Answers to questions will be posted immediately

2. Questions will be queued for posting immediately after moderation

3. Depending on the number of messages we receive, you could wait up to 24 hours for your message to appear. But be patient as posts will appear after passing our moderation.


MyLinks On Acalytica | Social Proof Widgets | Web Analytics | SEO Reports | Learn | Uptime Monitoring