Er, if I may, the question you asked is not phrased in a way that a non-Catholic Christian can answer it directly. You asked:
Using scripture alone show that Matthew wrote a gospel and that it is inspired.
Worded as follows the substance of your question can be answered:
How do you know that Scripture, e.g. the Gospel of Matthew, contains the words of God?
If you had asked it that way, I would say that Christians have the gift of "ears to hear." We know God's words when we hear them. Thus we know what is or is not Scripture. People who do not have that gift cannot hear Him and cannot discern the difference between the words of men and the words of God.
To him the porter openeth; and the sheep hear his voice: and he calleth his own sheep by name, and leadeth them out. And when he putteth forth his own sheep, he goeth before them, and the sheep follow him: for they know his voice. And a stranger will they not follow, but will flee from him: for they know not the voice of strangers. – John 10:3-5 It is the spirit that quickeneth; the flesh profiteth nothing: the words that I speak unto you, [they] are spirit, and [they] are life. - John 6:63
And in them is fulfilled the prophecy of Esaias, which saith, By hearing ye shall hear, and shall not understand; and seeing ye shall see, and shall not perceive:- Matt 13:14
Why do ye not understand my speech? [even] because ye cannot hear my word. – John 8:43
My sheep hear my voice, and I know them, and they follow me: - John 10:27
But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know [them], because they are spiritually discerned. - I Cor 2:13
And again:
And Moses called unto all Israel, and said unto them, Ye have seen all that the LORD did before your eyes in the land of Egypt unto Pharaoh, and unto all his servants, and unto all his land; The great temptations which thine eyes have seen, the signs, and those great miracles: Yet the LORD hath not given you an heart to perceive, and eyes to see, and ears to hear, unto this day. And I have led you forty years in the wilderness: your clothes are not waxen old upon you, and thy shoe is not waxen old upon thy foot. - Deuteronomy 29:2-5
And again:
But the anointing which ye have received of him abideth in you, and ye need not that any man teach you: but as the same anointing teacheth you of all things, and is truth, and is no lie, and even as it hath taught you, ye shall abide in him. - I John 2:27 For the word of God [is] quick, and powerful, and sharper than any twoedged sword, piercing even to the dividing asunder of soul and spirit, and of the joints and marrow, and [is] a discerner of the thoughts and intents of the heart. - Hebrews 4:12
We know God's words when we hear them. We don't need any creature to tell us. If and when we did need that, then we were not yet Christian.
But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his. - Romans 8:9 Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and [of] the Spirit, he cannot enter into the kingdom of God. That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. Marvel not that I said unto thee, Ye must be born again. The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit. - John 3:5-8
I suspect the persons who gathered, retained and passed along the Scriptures heard God's words also, i.e. knew their importance. But the glory for preserving His words goes to God not man.
The words of the LORD [are] pure words: [as] silver tried in a furnace of earth, purified seven times. Thou shalt keep them, O LORD, thou shalt preserve them from this generation for ever. – Psalms 12:6-7 So shall my word be that goeth forth out of my mouth: it shall not return unto me void, but it shall accomplish that which I please, and it shall prosper [in the thing] whereto I sent it. - Isaiah 55:11
God's Name is I AM.