There are numerous issues about the Gospels. Who wrote them and when they were written and if inspired how can they contradict each other. For example, Mark & Luke are not even named as apostles.
There is so much controversy that it would be impossible to explain here. The below is a small summary in relation to Matthew.
No mention of the author's name anywhere in this gospel. The earliest date (which is very arguable) that is given by Theologians is that it was written in the year 90 CE.
Papyrus (P64) if dated to 3 or 4 century according to palaeography, P104 second & P77 end of 2nd early third.
According to chapter 9, Matthew was a Palestinian Jew. The vast majority of Palestinian Jews in this period were illiterate – probably around 97%. The exceptions were urban elites. There is nothing to suggest that Matthew, the tax collector, was an urban elite who was highly educated. Albeit, not necessarily illiterate, he would have written in Aramaic, very unlikely to have known let alone written Greek in such manner.
Apart from the books written by the extremely highly literarily elite Josephus, we don’t have any literary books composed in written Greek by any Palestinian Jews of the first century.
Justin Martyr, writing around 150-60 CE, quotes verses from the Gospels, but does not indicate what the Gospels were named. these books are simply known, collectively, as the “Memoirs of the Apostles.”
Matthew is mentioned for the first time, in the writings of the church father and heresy-hunter Irenaeus, around 180-85 CE.
On the basis that Matthew was a similar age to Jesus who is said to have died at the age of 33, therefore Matthew was likely to be around 90 years old. Could he have really written it at the age of 90 and why wait so long?
If he was inspired by the HS why wait until after Marks Gospel who was not named in the bible as an apostle.
The writer of the Gospel of Matthew spoke about Matthew the publican, the disciple of Christ, in the form of absent, which means that Matthew the publican is not the author of the Gospel.
As Jesus passed by from there, he saw a man called Matthew sitting at the tax collection office. He said to him, “Follow me.” He got up and followed him. (Matthew 9:9)
"50 And Jesus said unto him, Friend, wherefore art thou come? Then came they, and laid hands on Jesus and took him. 51 And, behold, one of them which were with Jesus stretched out his hand, and drew his sword, and struck a servant of the high priest's, and smote off his ear." (Matthew 26:50)
"56 But all this was done, that the scriptures of the prophets might be fulfilled. Then all the disciples forsook him, and fled. 57 And they that had laid hold on Jesus led him away to Caiaphas the high priest, where the scribes and the elders were assembled. 58 But Peter followed him afar off unto the high priest's palace, and went in, and sat with the servants, to see the end." (Matthew 26:56-58)
"1 And when he had called unto him his twelve disciples, he gave them power against unclean spirits, (so they) to cast them out, and to heal all manner of sickness and all manner of disease. 2 Now the names of the twelve apostles are these; The first, Simon, who is called Peter, and Andrew his brother; James the son of Zebedee, and John his brother; 3 Philip, and Bartholomew; Thomas, and Matthew the publican; James the son of Alphaeus, and Lebbaeus, whose surname was Thaddaeus; 4 Simon the Canaanite, and Judas Iscariot, who also betrayed him. 5 These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not." (Matthew 10:1-5)
As he sat in the house, behold, many tax collectors and sinners came and sat down with Jesus and his disciples. (Matthew 9:10)
Matthew as like John does not mention;
"51 And it came to pass, while he blessed them, he was parted from them, and carried up into heaven." (Luke 24:51)
"19 So then after the Lord had spoken unto them, he was received up into heaven, and sat on the right hand of God." (Mark 16:19)
miracle of reviving Lazarus and the miracle of turning water into wine as mentioned by John
John (chapter 11 verse 1) & John (chapter 2 verse 2).
Difference with lineage of Matthew 1:1 & Luke - a number of issues with Matthew’s genealogy for another question.
"50 Jesus, when he had cried again with a loud voice, yielded up the ghost. 51 And, behold, the veil of the temple was rent in twain from the top to the bottom; and the earth did quake, and the rocks rent; 52 And the graves were opened; and many bodies of the saints which slept arose, 53 And came out of the graves after his resurrection, and went into the holy city, and appeared unto many. 54 Now when the centurion, and they that were with him, watching Jesus, saw the earthquake, and those things that were done, they feared greatly, saying, Truly this was the Son of God."
The bodies coming out of the grave never mentioned in any of the other Gospels. The Split of the temple not mentioned in John.
Papyrus 104 – versus 44 is missing.
Some examples of passages exclusive to Matthew (many more)
Birth of Jesus and Joseph’s dream (1:18-25)
Two blind men healed (9:27-31)
Peter walking on sea (14:28-31)
Disciples not to go to Gentiles but to lost sheep of Israel (10:5-8)
Women worshipping risen Jesus (28:9-10)
Many people heard from stories about Jesus and wrote about what they heard; such is evident from;
Gospel of Luke chapter 1 verse 1: "1 Forasmuch as many have taken in hand to set forth in order a declaration of those things which are most surely believed among us, 2 Even as THEY delivered them unto us, which from the beginning were eyewitnesses, and ministers of the word; 3 It seemed good to me also, having had perfect understanding of all things from the very first, to write unto thee in order, most excellent Theophilus, 4 That thou mightest know the certainty of those things, wherein thou hast been instructed."27
More evidence to say that Matthew the disciple of Jesus DID NOT write the Gospel.
You may also be interested in the Gospel of John - https://hermeneutics.stackexchange.com/a/63568/33268