My linear algebra professor likes to use theorems he expects us to prove later in his proofs of theorems in class. Well it's not like that because it makes sense but they're usually more of side notes. Today he asked us to prove this:(adsbygoogle = window.adsbygoogle || []).push({});

Given A and B are 2 by 2 matrices over the field R, prove that if A*B = I(2x2) where I(2x2) is the 2x2 identity matrix, B*A = I(2x2).

Now I filled up a good page just to prove this and ended up turning A and B into

[tex]A = \left( \begin{array}{cc} a1 & a2\\

a3 & a4 \end{array} \right) , B = \left( \begin{array}{cc} b1 & b2 \\ b3 & b4 \end{array} \right)[/tex]

into

[tex]A = \left( \begin{array}{cc} a1 & a2\\ a3 & a4 \end{array} \right) , B = \frac{1}{a1a4-a3a2} \left( \begin{array}{cc}a4 & -a2\\ -a3 & a1 \end{array} \right)[/tex]

and moving on from there it was pretty easy to prove that [tex]AB = BA = \left( \begin{array}{cc} 1 & 0\\0 & 1 \end{array} \right)[/tex]

Is there another way? I wasted so much time and my professor's proofs are usually so much more elegant. I can scan a copy of my work if you're still confused as to what I actually did (unbeautifully)

(Please excuse this post it will take awhile to get right I am learning LaTeX)

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# An interesting matrix identity problem

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