Thread safe singleton, locks

I have a Singleton class which is a thread safe, do I need to lock it methods?

        private static volatile JsonWriter instance;
private static final Object mutex = new Object();

ReentrantReadWriteLock lock = new ReentrantReadWriteLock();

private JsonWriter() {
}

public static JsonWriter getInstance() {
    JsonWriter result = instance;
    if (result == null) {
        synchronized (mutex) {
            result = instance;
            if (result == null) {
                instance = result = new JsonWriter();
            }
        }
    }
    return result;
}

do I need to lock each method like this to ensure thread safety?

    public void write(String filePath, ArrayNode content) throws IOException {
    lock.writeLock().lock();
    File file = new File(MASTER_DIR + "/" + filePath);
    mapper.writerWithDefaultPrettyPrinter().writeValue(Files.newOutputStream(file.toPath()), content);

}