Question - Matrix Multiplication and Equality Question
Solution:
We are given two \( n \times n \) matrices \( A \) and \( B \) such that \( BA = I \), where \( I \) is the \( n \times n \) identity matrix. We want to find out if \( AB = I \) as well.
To show that \( AB = I \), we have to use the fact that \( BA = I \). Since matrix multiplication is associative, we can use the following manipulation:
Let us multiply both sides of \( BA = I \) by \( B^{-1} \) (the inverse of \( B \)) from the left:
\( B^{-1}(BA) = B^{-1}I \)Since \( B^{-1}B = I \) and \( B^{-1}I = B^{-1} \), we have:
\( (B^{-1}B)A = B^{-1} \)Therefore:
\( IA = B^{-1} \)And since \( IA \) is just \( A \), we get:
\( A = B^{-1} \)If we multiply both sides of \( A = B^{-1} \) by \( B \) from the right, we get:
\( AB = B^{-1}B \)And since \( B^{-1}B = I \), finally, we have:
\( AB = I \)Thus, \( AB = I \) using the fact that \( A = B^{-1} \) which we found from \( BA = I \). Therefore, it does follow that \( AB = BA = I \).
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com