• The rise comes hot on the heels of Twitter's best day of trading since its IPO in 2013, with shares surging over 27% on Monday.
• "Through conversations with Elon in recent weeks, it became clear to us that he would bring great value to our Board," Twitter CEO Parag Agrawal said in a tweet.
• Musk replied saying he was looking forward to working with Agrawal and the Twitter board to make "significant improvements" to the platform.

Twitter shares closed up about 2% on Tuesday after Elon Musk was appointed to the company's board of directors.

The appointment, disclosed in a release filed with the SEC, comes after Musk took a 9.2% passive stake in the company, making him the largest shareholder. The stake is worth almost $3 billion.

Twitter's share price had climbed 7.32% to $53.63 at 9:02 a.m. ET Tuesday before markets opened. The rise came hot on the heels of Twitter's best day of trading since its IPO in 2013, with shares surging over 27% on Monday.

"Through conversations with Elon in recent weeks, it became clear to us that he would bring great value to our Board," Twitter CEO Parag Agrawal said in a tweet.

"He's both a passionate believer and intense critic of the service which is exactly what we need on Twitter, and in the boardroom, to make us stronger in the long-term," he added.

Musk replied saying he was looking forward to working with Agrawal and the Twitter board to make "significant improvements" to the platform.

On Monday, Musk asked Twitter users if they want an edit button to be added to the platform.

"Do you want an edit button?" Musk asked in a poll. At the time of writing, the poll had received 2,333,856 votes, with some 73.6% voting "yse" and 26.4% voting "on."

Agrawal retweeted Musk and urged people to "vote carefully" as the consequences "will be important."

On April 1, or April Fools' Day, Twitter tweeted a message through its official account saying that it was working on an "edit" button. When the company was asked if it was a joke, it said: "We cannot confirm or deny but we may edit our statement later."

While an edit button would allow Twitter users to fix grammatical errors, there are concerns that it would also enable them to edit the meaning of a tweet, which could be problematic if that tweet had gone viral and had millions of impressions.

In response, Meta CTO Andrew Bosworth claimed that he and his team solved the issue on Facebook "a long time ago" by including an indicator that it had been edited along with a change log.

Musk's appointment to the Twitter board comes after he said he was considering building a new social media platform. The Tesla CEO does not like that his tweets are scrutinized by regulators like the U.S. Securities and Exchange Commission.

"Given that Twitter serves as the de facto public town square, failing to adhere to free speech principles fundamentally undermines democracy," Musk tweeted March 26. "What should be done?"

Musk was subpoenaed by the SEC in November after he asked his Twitter followers if he should sell 10% of his Tesla stock — causing shares to fall.

Bilal Hafeez, CEO of financial market research firm Macro Hive, told CNBC's "Street Signs Europe" Tuesday that Musk has had issues with Twitter for a while, adding that he's unsatisfied with the firm's approach to free speech and product development.

"It appears that Elon does care about the direction of Twitter," Hafeez said. "The question is how active he will be in terms of the future direction of the company."

Hafeez added that Musk may try to bring in an outsider to run Twitter. Agrawal was formerly CTO before taking up his CEO role.

"More fundamentally, there is an issue of Twitter's revenue growth model," Hafeez said. "It does have aggressive targets for its daily active users, yet it's unlikely to be able to achieve those. Its revenue growth, while it did pick up in the pandemic, wasn't as large as many other platforms over the same period of time."