Boeing CEO Dennis Muilenburg said he would trust his own family in a 737 Max jet after testing new software in the wake of two deadly crashes. In October of 2018, Lion Air Flight 610 crashed, killing 189 people on board. Then in March of this year, Ethiopian Airlines Flight 302 crashed, killing 157 people.

After the second crash, Boeing 737 Max jets were grounded. Earlier this month, the company said it finished updating flight-control software indicated in both crashes.

But in an exclusive interview with incoming "CBS Evening News" incoming anchor Norah O'Donnell, Muilenburg acknowledged the incidents damaged public trust. A portion of their interview is below:

Dennis Muilenburg: I do personally apologize to the families. We feel terrible about these accidents, and we apologize for what happened, we are sorry for the loss of lives in both accidents, and that will never change. That will always be with us. I can tell you it affects me directly as a leader of this company, it's very difficult.

Norah O'Donnell: Did you ever consider resigning?

Muilenburg: No. It's important that I continue to lead the company and the fact that lives depend on the work we do, whether it's people flying on our commercial airplanes or military men and women around the world who use our defense products, that is a worthy mission.

The admission by the CEO of Boeing will be watched closely by people like Paul Njoroge, who lost his entire family in the Ethiopian Air crash. Njoroge said it was up to Boeing and "the others in charge" to save them.

O'Donnell: He's pointing the finger at Boeing. How does that affect you?

Muilenburg: Well you know I understand the feelings of all these, the loved ones and families who have been affected and I can't even claim to begin to comprehend how much it's impacted them and unfortunately I can't change what happened. I would if I could. But what I can commit to is our company is going to do everything possible to ensure safety going forward.

The common link between both crashes was the performance of a new flight control system called MCAS, which activated after it was fed erroneous information.

O'Donnell: The first crash, Flight 610 that crashed into the Java Sea about 12 minutes after takeoff killed 189 people on board. What's your understanding of what brought that flight down?

Muilenburg: Well first of all, I have to respect the fact that the investigation process is still going on. We know there was inaccurate sensor data that came into the airplane and there appeared to be a maintenance issue with that sensor. We know that the MCAS software was activated multiple times during that flight and in the end that added to the pilot workload.

O'Donnell: It was more than multiple times, it was two dozen times. The pilots were essentially in a tug of war with the plane, for control of the plane. That flight control system that you mentioned, the MCAS overroad the pilots more than two dozen times, and the pilots ultimately lost control and that plane essentially did a death dive into the ocean at 450 miles an hour. Could you imagine how terrifying that was for the people on board because it was jerking up and down.

Muilenburg: We examine every dimension of these accidents. Not to try to attribute fault or point fingers, but it's to understand... what happened.

Boeing admits it was a mistake in the software for a warning light, called an angle-of-attack disagree alert, that could have notified pilots and maintenance that there was a problem.

O'Donnell: But that light was supposed to be active on all 737 Max jets, and it was not. You knew this in 2017 and did not tell the FAA for 13 months. Why?

Muilenburg: The implementation of that software, we did not do it correctly. Our engineers discovered that.

O'Donnell: So you're fixing that now.

Muilenburg: We are fixing it now, and our communication on that was not what it should have been.

O'Donnell: Does Boeing have a credibility and transparency problem if they don't admit what were the mistakes in the past?

Muilenburg: No as I said, we clearly fell short and the implementation of this angle-of-attack disagree alert was a mistake, right, we did not implement it properly. We're confident in the fundamental safety of the airplane.

O'Donnell: You'd put your family on a 737 Max?

Muilenburg: Without any hesitation. Absolutely.

Watch more of O'Donnell's interview with Muilenburg on Thursday, May 30, 2019, on "CBS This Morning," 7 a.m. ET.