Matrix Multiplication and Equality Question
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.
<p>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:</p>
<p>Let us multiply both sides of \( BA = I \) by \( B^{-1} \) (the inverse of \( B \)) from the left:</p>
\( B^{-1}(BA) = B^{-1}I \)
<p>Since \( B^{-1}B = I \) and \( B^{-1}I = B^{-1} \), we have:</p>
\( (B^{-1}B)A = B^{-1} \)
<p>Therefore:</p>
\( IA = B^{-1} \)
<p>And since \( IA \) is just \( A \), we get:</p>
\( A = B^{-1} \)
<p>If we multiply both sides of \( A = B^{-1} \) by \( B \) from the right, we get:</p>
\( AB = B^{-1}B \)
<p>And since \( B^{-1}B = I \), finally, we have:</p>
\( AB = I \)
<p>Thus, \( AB = I \) using the fact that \( A = B^{-1} \) which we found from \( BA = I \). Therefore, it does follow that \( AB = BA = I \).</p>
