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Vector Algebra question

A = (1052)\begin{pmatrix} 1 & 0 \\-5 & 2 \end{pmatrix}

Find elementary matrices E1 and E2 such that E1E2A=IE_1 E_2 A = I
(edited 5 years ago)
Original post by E--
A = (1052)\begin{pmatrix} 1 & 0 \\-5 & 2 \end{pmatrix}

Find elementary matrices E1 and E2 such that E1E2A=IE_1 E_2 A = I


You've used this site enough to know that you should post any thoughts / attempts first.
Reply 2
Original post by RDKGames
You've used this site enough to know that you should post any thoughts / attempts first.

My thoughts
Elementary matrices are invertiable. And E1 and E2 must be a 2 x 2 matrices.
Original post by E--
My thoughts
Elementary matrices are invertiable. And E1 and E2 must be a 2 x 2 matrices.


Well, you're not wrong, but it isn't much to go on.

Notice that (E1E2)A=I(E_1E_2)A = I and the only way this is possible is if the product E1E2E_1E_2 is the inverse of AA. [We simply use the fact that A1A=IA^{-1}A = I here]

Hence E1E2=A1E_1 E_2 = A^{-1}. What is A1A^{-1} ?

If you can find two elementary matrices which multiply to give A1A^{-1} then you're done.
Original post by RDKGames
Well, you're not wrong, but it isn't much to go on.

Notice that (E1E2)A=I(E_1E_2)A = I and the only way this is possible is if the product E1E2E_1E_2 is the inverse of AA. [We simply use the fact that A1A=IA^{-1}A = I here]

Hence E1E2=A1E_1 E_2 = A^{-1}. What is A1A^{-1} ?

If you can find two elementary matrices which multiply to give A1A^{-1} then you're done.

I understand he hasn't given you much to go on, but I feel this is probably going to send the OP in the wrong direction; the intent of the question is clearly to use 2 elementary row operations to reduce A to the identity and then make E1,E2 the corresponding elementary matrices.

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