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I was trying to prove this -

Let $A$ be a lower triangular $n\times n$ matrix with nonzero entries on the diagonal. Show that $A$ is invertible and the inverse of $A$ is lower triangular.

[Hint: Explain why $A$ can be changed into $I$ using only row replacements and scaling. (Where are the pivots?) Also, explain why the row operations that reduce $A$ to $I$ change $I$ into a lower triangular matrix.]

I believe we can prove this by considering the matrix $[A|I]$, now we since $A$ is lower triangular we can first scale the pivot in the first row to 1 and then use it to remove all subsequent entries in first column below it then scale the pivot in column 2 and so on.

Clearly we can reduce the matrix to the identity matrix but at the same time I will have the same row operations so we'll be multiplying and adding multiples of the pivots to rows below it and generating a lower triangular matrix only since there will be no instance when we add a row to a row above it we'll only use the pivots to remove the entries in the column below it. So the inverse generated will be a lower triangular matrix only using the theorem which states that if a set of row operations reduce a matrix to I then they reduce I to the its inverse

Now this is logically correct as far as I know but this doesn't look much like a proof rather an intuitive argument to me and there is where I always get stuck. I get to a point where I know why it's true but don't know how to write it formally. So using this example can someone give tips and techniques to write it in a more "proof like" manner if I may say so

I guess it's been asked here but I'm not looking for a proof per say but a way to convert my intuition into a set of rigorous statements

Cameron Williams
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FoundABetterName
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For proofs, formulae or "lots of mathematical terms" are not necessary in general. There is a whole discipline for achieving exactly that: Proof without words, which is considered a high art ....

Andreas
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  • So would you say my argument is sufficient as a proof? – FoundABetterName Feb 05 '21 at 18:25
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    Well, you don't present a proof without words ... – Andreas Feb 05 '21 at 18:26
  • Yeah that's true but then I feel this doesn't answer my question on how to convert an intuitive logical argument to a formal proof – FoundABetterName Feb 05 '21 at 18:27
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    Depends on the audience what constitutes a "proof" for what it's worth. A proof without words can be a perfectly fine proof. A proof with many words can be a perfectly fine proof. Two mathematicians might look at the same proof and conclude differently about the validity of a proof. – Cameron Williams Feb 05 '21 at 18:27
  • @CameronWilliams I am not too sure that college professors especially at my level would be welcoming of some proofs which don't resemble proofs as we imagine them traditionally like my line of reasoning is correct but not something you would imagine to be in a book as a proof of the statement. So leaving aside the professors can this question be more focused on tackling the task at hand that is to convert the proof to something which resembles a textbookish proof? – FoundABetterName Feb 05 '21 at 18:29