[Alamo-Girl]

To the contrary, the way you phrased it was "begging the question" - a logical fallacy.

You asked:

Using scripture alone show that Matthew wrote a gospel and that it is inspired.

The "Matthew wrote" assumes without proof that Matthew wrote the gospel of Matthew, i.e. that it contains Matthew's words as compared to the words of someone else such as God.

And if Scripture alone were used in reply to the issue of "inspired" then the reply would also be a logical fallacy, i.e. circular reasoning.

And so, in the hopes this quandary was unintentional on your part, I rephrased your question for a more direct reply from a non-Catholic Christian like me:

How do you know that Scripture, e.g. the Gospel of Matthew, contains the words of God?

If that question does not satisfy you then there is nothing more I can add because I choose to not answer a baited question.

[vladimir998]

Either sola scriptura works or it doesn’t. You’re proving it doesn’t.

[lastchance]

How would you explain such to a person who did not have even a Pagan understanding of faith? You are faced with someone who is convinced that radical Atheism has the best answers to explain the world. How do you get them to listen to you and at least consider that the Bible is inspired? How do you introduce them to Christ?

They have no common language of faith so how do you reach them?