As the protagonist in The Rules of Survival, Matthew is also the oldest child in the Walsh household. Because of this, he's had to put up with Nikki's lies and abuse for the longest, and feels pretty jaded about the way life works. When Murdoch breaks up with Nikki, Matthew just assumes that he's going to abandon the kids and that he doesn't care about what will happen to them either:
It surprised me what I felt as I looked at Murdoch, who stood like a statue in her embrace, before he reached down and simply forced her away from himself, pried her off as if she were a piece of garbage, and stepped decisively away.
It felt as if he was treating us—you, me, and Callie—like garbage, too. (11.35-36)
Matthew has a hard time trusting adults because they've never done him any good. No one has ever looked in and saved him and his siblings from their terrible situation, not even his own dad. Even when Murdoch convinces the adults in their lives to start paying attention and taking responsibility for the kids and their wellbeing, Matthew doesn't fully trust their motives or follow through. He even admits that after years of living with Aunt Bobbie—who treats them like her own children—he still doesn't completely trust her:
But they were sisters. And this, Emmy, may ultimately be why, though I adore Aunt Bobbie, I will never quite trust her completely.
Please, Emmy, never tell her. I know it's wrong of me. (31.11-12)
Matthew can't help it if he's cynical and thinks that he's the only person he can rely on. His early experiences have taught him just that, time and time again.
As the oldest child, Matthew also feels like it's his burden and job to take care of his little sisters, and to protect them from the wrath of Nikki. He's done this for them their whole lives:
It was hard to figure out what would be the safest thing to do, for all three of us, all the time. But it was my job. (1.13)
Because Matthew is the oldest kid, he feels like he has to do all of the protecting in the household, because Callie and Emmy can't fend for themselves. In doing so, he pretty much has to throw his childhood away. It's only when he moves in with Aunt Bobbie that Matthew is able to relax in his role as the protector… and is able to start living his own life.
Even though he's already a teenager the first time that he sees Murdoch—and therefor has survived years of Nikki's abuse—Matthew can't help but be drawn to the man. He sees Murdoch as their eventual savior, since he so easily jumps in to help the little boy at the grocery store:
But Murdoch was still looking straight at him, and I knew—you could feel it vibrating in the air—that even though Murdoch had said he wouldn't hit him, he wanted to. He wanted to hurt him.
I liked him for that. No, Emmy, I loved him for that. Immediately. (1.25-26)
Matthew gravitates toward Murdoch because he feels like he's the kind of adult who will actually do something to save them all. And it's true—when Matthew reaches out to Murdoch for help, he's the first adult who comes through and acts.
Matthew undergoes a huge transformation from the beginning of the book to the end. At first, he's just a kid resigned to his fate—he's convinced that his whole life will consist of evading Nikki's cruelty and violence. Then, when he meets Murdoch, he becomes obsessed with the idea that Murdoch could save them all, that maybe he'll be their savior and take them out of this awful home.
But eventually, as Matthew gets older, he comes to realize that he has a lot more power than he initially believes. He can testify against Nikki; he can stay away from her and ignore her screams and shouts; he can even save Emmy from her. By the very end, Matthew has grown up enough to look back and reflect on his time with Nikki. He realizes that he's written this whole story down so that he could sort through his own feelings:
So. Emmy. Little sister. You're never going to read this, are you? I'm never going to give it to you. I didn't write it for you. I wrote it for me.
I wrote it to work my way through the story of what formed me. (E1-2)
And now that he's developed the maturity to work through his complicated emotions, Matthew can finally move forward.