History
Most of Matsuda's life up to the point where he joins the Japanese police force is unknown. We can gather from things that he says throughout the series that he was always a weak person, and he admits that he only joined the police force because he had a connection. Maybe he chose it because he dislikes slow and steady work, but they took him in, and I assume that Matsuda's training is what allowed him to become such a good shot with a gun. However, he was still one of the youngest members on the force, and was still lacking in experience when Kira first appeared.
Back when Light Yagami was just starting out with the Death Note, the world's police forces were also just starting their investigation into the "mysterious criminal deaths," but all that seemed to turn up were dead ends. Matsuda happened to be sitting in on his first ICPO (otherwise known as Interpol) meeting when the topic was brought up, and the Chief of Police, Soichirou Yagami, had to explain to him who L was as the great detective's voice began explaining the situation. Both policemen were surprised when L asked for Japan's cooperation, in particular, but they pledged him their full support and all available members of the force were assigned to the case.
L's first order of business was to set up a trap, though the police force wasn't aware of it being so, by having an inmate on death row pretend to be him for a special live broadcast. Kira, or Light, took the bait and killed the man on TV, informing L and the police force that he could kill from a distance, and that he needed at least a face and a name.
During the following weeks, the police force was asked to give a daily update of the situation, and one day Matsuda commented on the significant decrease of criminal activity. The room fell silent, and was left wondering if he should have stayed silent to begin with; but when he asked Soichirou that question later, he was told to keep the facts coming no matter what they were, especially on the more delicate matters like the decline of criminal activity.
After a few weeks, it came out that L had had the FBI investigating members of the Japanese police force in secret, and that all their agents had died simultaneously. Not only did this weaken the police force's trust in L, but it also made many of them worry about whether or not Kira would kill them. Then Soichirou asked all of them to think about what they had at stake, and to leave the Kira investigation. Most of the police force did, and only five other people stayed with Soichirou to continue investigating. Matsuda was one of those people, along with Mogi, Aizawa, Ukita, and Ide. L said that with himself and Watari, there were seven, but the policemen weren't sure. L asked them to go and decide whether or not they would trust him enough to work with him. Ide said that he didn't trust L, and left, but the others stayed with Soichirou. They reported their decision to L, who told them what hotel he was staying in and a specific way he wanted them to arrive.
When the policemen finally met L face-to-face, they were surprised, probably because he didn't look as they thought a "great detective" should. But they proceeded to introduce themselves. L lifted his hand as though it were a gun and pretended to shoot them, telling them that if he was Kira he could have killed them there and asked that they try not to get themselves killed, then proceeded to describe the way things stood and state that justice would win in the end. Despite any previous doubts that he might not have actually been L, this proved his identity to the rest of the task force.
There were a few things that Matsuda didn't like from that point, such as L's obvious exemption from his own rules, like turning off cell phones, but he was still happy to be working on the Kira case and stuck with it, even though L had them working until they dropped asleep from exhaustion. For a few weeks, they made some small progress in narrowing the suspects down to Police Director's family and Chief Yagami's family, and L even had bugs and cameras installed in their houses, but they didn't find any suspicious behavior in either and were sent back to square one. At that point, L followed the Chief's son, Light, into college to investigate him up close, while leaving the rest of the task force to look into other aspects of the case in his absence.
The arrival of a second Kira was where things began to get more difficult for the task force, and Light was brought in as a consultant on the case, lifting the task force's spirits a bit now that there were two geniuses instead of one. A few video messages were exchanged between the task force, under the disguise of being the real Kira, and the second Kira, and eventually they received a page from a diary listing several future dates that the second Kira wanted publicized. The team took this to mean that on one of those days, and at one of the locations, the two Kiras were going to meet up.
In order to try and figure out where the two Kiras were going to meet, Matsuda went to Aoyama with Light to see if anything would happen, and the other task force members took several of the other dated clues. In order to provide some cover for them, Light had friends of his from college come with them, and had Matsuda pose as his cousin. They didn't notice anything out of the ordinary, but after all the dates on the journal page had passed, there was another message from the second Kira stating that they'd found the first Kira. Attached to this was hair, small fabric threads, and other evidence that the task force immediately began examining.
It didn't take them long to come up with a suspect: Amane Misa, a model and actress, and Light's new girlfriend. When they went to arrest her under suspicion of being the second Kira, they had to follow her to Light's college, but she was blindfolded and arrested from behind, then taken to a special holding cell.
Misa was kept for about a week when Light called and admitted that he was afraid he might be Kira. The rest of the task force tried to convince him that he wasn't, that after watching him there hadn't been any suspicious movements on his behalf, but to be certain he insisted on being held and watched 24/7. L agreed to it, and though the rest of the task force was uneasy about it, they took him in, and even put Chief Yagami in a cell at the man's own request. For several weeks, it seemed as though Kira had stopped killing, even though Light had a change of heart during his imprisonment and insisted that he knew he wasn't Kira. The rest of the task force was baffled by this.
When Kira's killings began to happen again, Matsuda rushed to inform the Chief, because it meant that Light couldn't possibly be Kira. However, L stopped him before he could let Light know, and instead decided to use this moment to test Light again, but Light continued to insist that he wasn't Kira. In order to solve the problem, L had the team act as though Light and Misa had been found guilty and would be executed, and then the Chief took a detour and acted as though he were going to shoot Light right there and then kill himself. The idea was that Kira would even kill a parent, but the Chief didn't die, and L agreed to release the two. The entire investigation was also moved to a hotel that L had had built specifically for it.
L wanted to keep Misa under surveillance after he released her, since she still had work, and Matsuda, eager to prove that he was useful, became her manager and escorted her to the various sets she was working on. This also involved little more than him being an errand boy and had him bringing her sweets and coffee and whatever else she wanted, and he had to keep up the role of a super-excited manager no matter how tired he was from working on the case, but I think he was happy to be useful up until that point.
During this time, the task force members were informed that they would have to choose whether to keep their jobs as policemen or stay on the case, as Kira was intimidating the police force to stop investigating. Matsuda happily and quickly chose to stay on the Kira case alongside Chief Yagami, who had already made his own choice, but Aizawa ended up leaving. It was a blow losing one more person, as the team was already short-staffed as it was, but they managed to get along without him.
Eventually, the task force noticed that the current Kira deaths were more beneficial to a certain company called Yotsuba, although they still needed to brainstorm how to get in and investigate if Kira was there or killing people on their behalf. But Matsuda was even more determined than usual to prove he could be useful, after having been put down and used as the coffee boy the last week or two, and went to Yotsuba on his own while Misa was on set for a movie. He managed to infiltrate it successfully, and overheard two of the people talking about secret meetings at the end of the week, and managed to catch that they were going higher up in the building rather than down to leave, so he took the stairs and followed them all the way up.
He managed to follow them to the meeting room they were in, and tried to listen in through the door. It took him a while, but he managed to hear the words "Kira" and "kill". He was so excited about hearing the evidence, however, that he didn't realize that two people were getting closer until they opened the door on him and he fell through the doorway to find eight of Yotsuba's top people in the meeting. He was shocked, but to his credit, he didn't let anything slip about the investigation. First, he quickly and secretly let L know where he was via a transmitter in his belt that contacted Watari, and then proceeded to insist that he was Tarou Matsui and was there to convince them to hire Misa for their commercials. Two of the Yotsuba people escorted him to another room to keep an eye on him while the rest of them stayed behind to talk. While he was being watched, Matsuda got a call from L, who managed to get the information he needed that Matsuda was in trouble and alone without letting the Yotsuba people know what was going on.
Matsuda had Misa come over when she was done shooting the movie for the day, in order to prove to Yotsuba that he was really her manager, and found out through her that L had a plan to invite Yotsuba back to headquarters where he could watch what was going on. Except in a more subtle way. The Yotsuba people followed them under the pretense of having a drink, when they also wanted to keep an eye on him since they thought he'd overheard their plans.
Matsuda was able to get away from them long enough to call L and inform him of what he'd heard, at which point L confirmed that they would probably want him killed. The solution to keep Matsuda alive involved him purposely falling off of the side of the building and faking his own death. This idea surprised, and probably also scared, Matsuda, but he trusted that L would take care of him and went along with the idea. He pretended to be drunk and tried provide the Yotsuba people with "entertainment" by doing handstands on the edge of the balcony, and purposefully slipped when he was sure that there was someone on the floor underneath him to catch him. He fell on a mattress, and at the same time a sandbag was thrown over the edge to make the Yotsuba people think he had hit the ground below. One of the other people L had hired, a con artist named Aiber, was dressed up as Matsuda in order to further encourage those beliefs, and they even went as far as to use an ambulance to get Aiber out of there safely. The plan worked; to Yotsuba, "Tarou Matsui" was dead.
After that, he took on a much more private role, while Mogi took his place as the excited manager. He was supposed to be a dead man, so he stayed at headquarters and went over paperwork. At the same time, though, his information allowed them to get inside Yotsuba, and the company even hired Misa. The task force bugged the meeting room, through the help of a thief that L had hired named Wedy, and they proceeded to make contact with one of the men in an attempt to narrow down the suspects. It was made even easier for them when one of the eight, a man named Hatori, was killed for wanting to stop having the meetings, which further proved that Kira had to be one of the remaining seven.
Eventually one of the seven, a man named Higuchi confessed privately to Misa (under the belief that she was the second Kira) that he was Kira, unaware that she was recording it, and the task force began to form a plan to catch him. They needed to corner him, and the best way was to use someone who might actually know the truth. Someone like Matsuda, who was supposed to be dead. L told him from the beginning that it was possible he could die, but Matsuda agreed to go through with it anyway.
The task force used Sakura TV to broadcast a three-hour long special that said it would reveal at the end who Kira was, in which Matsuda was supposed to buy time while Higuchi panicked. He was set up behind a witness screen, and near the beginning of the special, they purposefully let it fall so that Higuchi could see his face. Despite this, Matsuda continued to buy time while Higuchi went around trying to find his name, and they eventually switched him and the interviewer when Higuchi began to make his way toward the TV station in a last act of desperation. During this time, he proved that he could kill without needing a name, similar to the second Kira, and the task force made sure to keep their faces covered. A trap was waiting for him, but he managed to get outside, and the simple arrest turned into a car chase.
Aizawa and Ide showed up to help block off his route, along with the rest of the police force, and through everyone's combined efforts Higuchi was caught. He told them that he could kill people simply by writing names in a notebook, and while everyone was skeptical at first, they confiscated the notebook. Immediately after doing so, whoever touched it saw a monster, which was in reality an actual shinigami. They tried to ignore it and continue with the process of arresting Higuchi, but he had a heart attack before they could even get him in a police car. Everyone was even more confused by this turn of events, but decided to return to headquarters with the notebook to try and sort things out. The shinigami, Rem, went with them, but wasn't very helpful when it came to answering questions.
It didn't take long for another Kira to crop up and begin killing criminals again, which left the task force completely stumped. L proposed that they test out one of the rules in the notebook which would solve the whole case if proven false. Before they could go any further with that, though, Watari was killed. Before he died, he managed to hit a switch which began deleting all the Kira case data on the computers. L began to explain that he told Watari to do that if anything happened to him, and knew that Rem was up to something, but he wasn't able to finish explaining what that was. Rem had written his name in the notebook and given him a heart attack. By the time the task force got him to the hospital, it was too late. They gave him, and probably Watari, a proper burial, even though the task force were the only ones there.
Light took over L's position after the rest of the task force agreed that no one else could do it, and told them that he wouldn't be the same dangerous methods that L had used. Five years passed, and still nothing new appeared for the Kira case. The remaining six Yotsuba members, as well as Aiber and Wedy, had been killed shortly after L died. The world was becoming an even more dangerous place for those who opposed Kira, because those who agreed with him or thought he was a god were having their voices heard through television and other media. The detectives worked on the case as hard as they could from an apartment that was apparently being rented for that purpose, and Ide even came back to help them, but they were as lost as they'd been before Yotsuba. It's a bit surprising that Matsuda stuck with it for that long, since it's in his nature to dislike slow and steady work, which is what the Kira case had turned into. After L's death he probably felt he had to help catch him for the detective's sake, even though they were all even more sure that their lives were on the line.
One day while Matsuda, the Chief, and Light were at the Yagami residence discussing the case, they got a call from Aizawa that the current director had been abducted, and that the captors, led by a man named Mello, wanted to exchange the director for the Death Note. There was also an agent from the FBI looking to take the Death Note from the task force, claiming that they were untrustworthy. The director ended up dead, which the captors claimed was their doing when Light said it was probably Kira, but instead they still wanted to make the trade. This time, the hostage was Light's sister Sayu. Light tried to enlist the aid of the FBI, and ended up talking to someone named "N", who was the head of another Kira investigation known as the SPK. In reality, he was a boy named Near, raised to be L's successor. Light managed to get Near to give them some help through satellite surveillance while letting the Chief go through with the trade. There wasn't much that the rest of the task force could do except watch and try to find a chance to make a move.
In the end, it didn't do any good, and the notebook was lost. Sayu and the Chief were recovered safely, but Sayu was in such shock that she didn't respond to any stimuli, and was wheelchair bound. After this, the Yagami family, aside from Light, moved out into a house away from the rest of society. The Chief didn't come back to headquarters, and the task force was again short a valuable person.
A week or two later, the Chief got a call from Kira, who wanted to help the team to get the Death Note back from criminal hands. To prove that they were Kira, they sent the team their own Death Note. There was another shinigami named Ryuk following this one, and it wasn't hard to figure out that someone on the team was supposed to make the deal to get the shinigami's eyes in order to see the names of Mello and anyone else who got in their way. Matsuda offered to make the deal, but the Chief insisted that he be allowed to do it, to make up for losing the notebook. No one objected to it, and the team made their preparations to storm Mello's hideout.
The Chief made the deal as the task force waited outside the hideout, and when he signaled it, they rushed the building. There were only a few people to deal with other than Mello, and Matsuda was easily able to incapacitate the ones who had the Death Note. However, Light wouldn't let them pull out after securing the notebook, and said that they had to find Mello, so the task force continued to search the building. An explosion separated them from the Chief, who managed to find Mello, and the sound of gunshots let the rest of the team find them. By the time they found the Chief and Mello, though, the Chief had been shot and Mello had his hand on another trigger for a bomb. Instead of setting it down, he pushed it, setting off explosions throughout the rest of the building.
The team was rushed to the hospital afterwards, and Light met them their. Matsuda and the others just needed some light treatment for their wounds, but the Chief was in critical condition. He still had the shinigami eyes, and confirmed that since he could see Light's lifespan, he wasn't really Kira, and then died. The rest of the team was distraught, but they still had jobs to do. They all met up on the roof after saying their goodbyes to the Chief and gave the notebook that they'd retrieved from Mello back to the shinigami it belonged to, one named Shidou.
They probably stayed at the hospital for a short period of time to help Light and his family arrange a funeral for the Chief, while sending one person back to watch headquarters and rotating who was on duty there. Shortly after that, though, when they were all back in headquarters, the United States' president held a press conference and announced that they wouldn't oppose Kira any longer. The team was shocked and furious at hearing this, but Matsuda decided to use this moment to talk to them about something he'd been thinking: whether Kira was really as evil as they were making him out to be. Matsuda admitted that he knew that it was his job to catch Kira, and wanted to, but he could also understand what Kira's sympathizers were feeling, since he was a weaker person than the rest of the task force members. Aizawa and Ide seemed surprised that he'd be saying such a thing, but Light said he knew how Matsuda was feeling and went on to explain how he thought Kira saw himself.
Not long after that, they got a call from Near saying that Mello had been caught, but escaped. Near used this conversation with Light to discover that there was a shinigami there with them, and then proceeded to tell the rest of the task force that if there was ever a doubt in their minds that L (or Light) was Kira, then they could contact him, and proceeded to give them a number to call. Shortly after this, Near's hideout in America was attacked by Kira supporters, though he made it out safely and contacted them again. Despite this, Matsuda never doubted that Light wasn't Kira, so he never used the number that Near gave them, like Aizawa did.
In an attempt to follow another lead into Kira, Light suggested that they use his new spokesperson, Takada Kiyomi, his old girlfriend from college. He managed to arrange a meeting with her, and the team set up cameras and bugs. During the meeting, though, Kira contacted her, and Light had removed the bugs and cameras. Things turned out okay, and there were several more meetings after that, while the task force listened in through just bugs. For Matsuda, this was also a source of entertainment, as he became interested in whether Light would end up choosing Takada or Misa, since he was essentially dating them both at the same time.
Eventually, Misa and Mogi were taken into custody by Near, who suggested both teams meet face-to-face. Light thought that would be a good idea, so Near suggested that they meet on January 28, 2013 at a dock warehouse known as the Yellow Box. Two days before that happened, though, Takada was kidnapped by Mello. The team watched the news in hopes of hearing new, but Takada eventually contacted Light herself, having managed to "get away" from her kidnapper. (In reality, she killed him with a piece of the Death Note.) Matsuda was told to stay and watch headquarters while the others went out to find her, but they came back with the bad news that the truck she had been in had crashed into an abandoned church, which then caught on fire. Takada hadn't made it out alive.
Despite Takada's death, the task force and the SPK were still set to meet on January 28 at the warehouse. Each group spent the few days left before the meeting to prepare in their own ways, and on that day, the groups finally met face to face. However, Near happened to be wearing a mask of L's face at the time and refused to take it off until 30 minutes had passed, to ensure that no one in either group died. The task force thought it a bit excessive, but both sides agreed and waited in tense silence.
When the half-hour was up, Near finally removed his mask, but once again asked them to wait once more before they got down to the business of who was Kira. He explained that they were waiting for someone to come and write everyone's names down in the notebook, and both Matsuda and Ide began to desperately ask why that was necessary. But they didn't have much time to let that sink in, as Near pointed out that the current Kira, a prosecutor named Mikami, was waiting right outside the door. As Mikami wrote their names in the book (making his declarations of "sakujo," or some variation of "eliminate"), Matsuda tried to draw his gun to defend himself, but found two of the SPK's guns aimed at him while Near assured them that they wouldn't die.
Once Mikami was finished writing their names down, Light and Near asked him to come inside. He did, at Light's insistence, and also counted off how much time was left since he'd written the first name. When he counted the last second, Light announced "Near, it's my win" and the teams braced themselves for what they thought was their impending deaths. But after a long moment they relaxed a bit and glanced around, and Matsuda announced what the entire task force, at least, was thinking: "We didn't die."
Lester and Gevanni, two of Near's people, grabbed Mikami and brought them the notebook, at which point he showed everyone whose names were written. Light's was absent. Light lost his normal composure at that point, and began shouting that it was a trap and claiming that he didn't know Mikami, who cried out in response to being forsaken. Aizawa pointed out that he'd said it was his win, which shut Light up, while the rest of the task force was at a loss for words. Matsuda was in shock from the evidence built up against the man he'd trusted for all these years, and fell to his knees and just asked Why while the rest of the task force began arresting Light, who broke away and ran over to the nearby wall. Everyone listened while Near explained how the notebook Mikami had brought was a fake, and even revealed that he'd been able to see Ryuk since they'd arrived.
The task force had listened quietly during the explanation, but shortly after Near finished his explanation, Light began to laugh insanely. The task force gave him horrified stares as he continued to laugh and then admitted to being Kira. Then Light launched into a speech attempting to justify his reasons for using the Death Note when he originally got it. Throughout the speech, Matsuda's shock and horror began to change into anger as he realized that everything they'd done for the past six years, everything L and Soichirou had done, had been for nothing, but he remained on his knees on the ground.
Near called Light nothing more than a crazy mass murderer, which caused a tense silence, but eventually Light broke it by asking if the notebooks that were there were real. He walked a short distance away from the others, elaborating his idea, but eventually showed his hand: it had been a distraction so that he could write down their names on the piece of the Death Note hidden in his watch, starting with Near. At that point, Matsuda finally let his anger get the best of him and reacted almost instantly by drawing his gun and shooting Light once in the hand, preventing him from finishing Near's name and shocking the rest of the task force.
Matsuda then proceeded to ask Light what his father had died for if he had been Kira this whole time. Light attempted to twist his words, thinking Matsuda would understand what he'd been trying to achieve, but it only served to make him more upset, causing tears to run down his face. When Light tried again to finish Near's name with his blood, Matsuda shot him four more times, but Aizawa, Ide and Mogi all jumped forward to stop him from putting the final bullet in Light's head. When they finally let him go, he'd calmed down, but had sunk into a light version of apathy and simply sat there while Light was writhing and bleeding on the warehouse floor.
Mikami eventually stabbed himself out of agony at realizing that his god was a simple human murderer and began gushing blood, unintentionally drawing everyone's attention away from Light, who scrambled up and ran. Matsuda snapped out of his apathy and called out, but Light just kept on running. Near tried to keep the task force from chasing after him, but Aizawa said wouldn't listen to his orders and the task force spread out to look for Light, since it was obvious he couldn't go far. It's assumed from the end of the anime that they eventually found him dead on the steps of another warehouse building that he had run into. This would have undoubtedly upset Matsuda quite a bit until some form of examination showed that he had a heart attack before bleeding to death from the bullet wounds.
Note: Matsuda has been taken right after Chief Yagami's death and before any of the following events. I just tl;dr.
Personality
More than anything else, Matsuda is impulsive and unpredictable. Matsuda is relatively naïve compared to the rest of the task force, and a major optimist, as seen by how completely he believes that Light can't be Kira until he hears the truth from Light himself; but he's not a complete pushover or idiot. He's really the only character who thinks about the topics at hand, and while characters like Light and L have chosen their sides of justice and never question what they're doing, Matsuda instead can understand both sides of things, and wonders more than once if what they're doing is right or if Kira is right. Matsuda also has an inferiority complex that is rivaled only by Mello's. He knows where his limits end, especially when surrounded by geniuses, but he still feels the need to prove that he's useful. These are usually impulsive plans that don't end well, as seen when he makes the decision to infiltrate Yotsuba on his own, unaware that L was making a plan to do just that, and ended up getting himself caught. He also knew it would be beyond his power to get out of that situation, and turned to L for a solution.
Matsuda is also generally recognized at the gentlest of the task force, but that doesn't mean he won't do what's necessary when the time comes. We see two good examples of this throughout the series. The first is during the raid on Mello's hideout, when he shoots the two of Mello's remaining subordinates only enough to stop them from shooting. The second is probably where we see Matsuda's less-gentle side best, when he finally learns the truth about Light. At first, he simply shot at Light's hand to keep him from writing on the hidden piece of the note; but after Light tried to convince him to shoot the others, he flew into such a rage that he shot Light multiple times, and the rest of the task force had to stop him from putting a final bullet in his head. Even after all that, his rage was more on behalf the people that Light had gotten killed, such as the Chief. For the most part, though, Matsuda is a happy, kind person.
Physical Description
Matsuda has a lighter build than many of his fellow policemen, standing at 5'9" and weighing 139 lbs. If eyes are the window to the soul, then Matsuda's an open book; his brown eyes don't know how to hold back any emotion. He also has black hair that's usually parted to the left. At the point I'm taking him from, it'll be cut just above his ears, although he has worn it longer and shaggier in the past. His clothing preferences change with whatever's fashionable at the time, although when in a suit he tends to dress a bit loosely, likely for comfort.
Powers
Matsuda has no special powers. On the non-magical scale, he's an accurate shot with a gun, and is familiar with basic police training (self-defense, etc).