It's complicated. The phrase "parts of Vietnam was matriarchy and installing a patriarchy government created tension and difficulty" is technically true, but there are caveats to this.

This assumption that the Vietnamese society being matriarchy causing trouble for the Chinese rule is based on the perspectives of later Chinese, Vietnamese, and American historians in the 60-70s. All 3 groups had political motivations to do so. The Americans (and the Vietnamese Americans) promoted the image of a Vietnamese woman, whose proper inheritor is the RVN and who is always in opposition to the northern men (in their metaphor, the "Chinese puppet" DRV), thus justifying their continuing support for the RVN. You can also see variations of this propaganda in the Miss Saigon musical and in some of Nixon's speeches. On the other side of the coin, the Chinese and Vietnamese at the time wanted to promote feminism and mobilize women to participate in the country-building/war effort respectively, and they would use Engels' theory that all societies started off as matriarchy to push their propaganda. The Vietnamese side also had a vexed interest in mobilizing Vietnamese women for wars and not just for paramilitary, so the Trưng sisters rebellion and the Lady Triệu rebellion were also used as proofs that Vietnamese women had half the responsibility to "hold up the sky" too. This narrative of an originally matriarchal society of the Vietnamese side continued throughout the Sino-Vietnamese war and only started to change around the 90s, when the need to mobilize women in addition to men for wars had largely ended. It was then that Vietnamese historians started to look at the Houhanshu and archeological records.

So what does the actual historical and archeological records say about Vietnamese society before and during the Han rule? Well, it's complicated.

The Houhanshu and later Vietnamese records (which took most of its information from Houhanshu + oral stories + editing) depicted the Vietnamese society through the family structure of the Trưng sisters as a patriarchy but matrilineal before the official start of the Sinicization campaign. Later Sinicization campaigns also depicted the "Nanman" in northern Vietnam as "not having the proper Confician family structure" (aka patriarchy) and women marrying many times. However, the Houhanshu was written some 400 years after the events of the Trưng sisters rebellion, by authors who strictly followed the patriarchy system, and the later Vietnamese sources copied off the Houhanshu, so take that as you will.

The 14th century Vietnamese source Lĩnh Nam Chích Quái and the 15th century Vietnamese source Đại Việt Sử Ký Toàn Thư both recorded the Trưng sisters' family structure as being completely patriarchy. However, they did copy that part from the Houhanshu. They also recorded the Vietnamese society beginning as a patriarchal society with the Hồng Bàng myth.

However, a curious detail in that Hồng Bàng myth is that the ancestors of the Vietnamese were depicted to have followed their mother to the mountain instead of their father to the sea. Later researches on the myth of the Mường people (another descendant group of the ancient Vietnamese, but a lot less Sinicized compared to the majority Kinh people) also depicted their creation myth as a mother goddess who created the world. In the modern day, many Mường groups are still matrilineal. The Kinh group themselves, though more Sinicized than the Mường group, has historically had higher position for women in their society compared to Chinese women even after Sinicization (for more information, see Hồng Đức code of the Lê dynasty in the 15th century). The Kinh group also still has the Mother Goddesses worship (which was recognized as a UNESCO intangible heritage) in the modern day. Unearthed daggers and statues of the pre-Chinese rule era also depicted mother goddess decoration. There are many remains of the matriarchal society still existing and thriving in the modern patriarchal society of the majority group Vietnamese Kinh.

Overall, the modern official and commonly accept sentiment is that Vietnamese society did start as a matriarchal society. It's just that it started transitioning to a patriarchal society even before the Chinese rule. The 2 Vietnamese sources above recorded that there were already 18 generations of male kings before contact with the Chinese. Even if we dismiss that part as myths, the existance of the very male king An Dương Vương was attested by both contemporary Chinese sources and archeology, especially when his descendant - Tây Vu Vương, also led a rebellion against the Han, was most definitely a descendant of An Dương Vương (they used the same tittle), and definitely existed as a male rebellion leader as recorded by Chinese history. Even if we assume that An Dương Vương's patriarchal Âu Việt culture had minimal impact on the wider Lạc Việt culture of northern Vietnam, An Dương Vương's existence at the very least confirmed that Vietnamese society had already started to transition to a patriarchal society before the Han rule, albeit mosaic and heterogeneous like it still is in the modern day.

So your student writing that "part of Vietnam was matriarchal" was technically correct, though in the grand scheme of thing, the majority of the population in northern Vietnam, esp the ones living near the political capital in Cổ Loa, was already patriarchal with matrilineal mixed in.

Now for the second part of your student's phrasing. I highly suspect that your student was influenced by the traditional Confucian Chinese narrative that to be considered Hua (in the meaning of civilized, and thus incorporated into the Huaxia empire), the Yi (meaning "barbarian") must adopt all the Huaxia customs, which included everything from hair style and clothes to family structure. As a result, Chinese policy makers tended to view any different custom of the "Yi" as agents of chaos and instability and needed to be stomped out. In 2nd AD, after the failed rebellion of the Trưng sisters, the Han government started to impose their culture, patriarchy, and other customs onto the locals in northern Vietnam with this belief in mind. Most of the modern Vietnamese words for family originated from the Old and Early Middle Chinese of this period. After 2nd AD, you can say that the majority of the Vietnamese society was definitely patriarchal and patrilianeal, albeit still mosaic like in the modern day.

However, that the differences in family structure contributes to tension in northern Vietnam under Han rule is still a theory and a belief held by the Han rulers. Before the Trưng sisters, Vietnamese society was already transitioning on its own, and there was the male Tây Vu Vương. After the Trưng sisters and the finished transition to patriarchy for the majority, there were still rebellions in northern Vietnam, led by both women and men. Except for a small window of stability during Sĩ Nhiếp's (de facto autonomous) reign, the intensity and frequency of rebellions steadily increased until the 10th century, when it finally broke free for good. While it's impossible to prove that the differences in family structure did not contribute to the tension and instability of the Han rule in northern Vietnam, all historical events before and after the Trưng sisters point to the fact that there were many more important reasons at play than just a difference in family structure. Like geography and the Chinese central governments' tendency to send people it disliked to the rich frontier province of Jiaozhi.

So yeah, the second part of your student's phrasing is true, on a technicality. It's just not the most important reasons as to the why.