I agree with Smoothie queen, there is usually some reason for why they feel they can’t be wrong.
Maybe it makes them feel worthless, less like an authority figure, more like people won’t look to them for help because they were wrong. Also, don’t forget denial. I knew of people who said they could never be wrong ever. They were always right. (Which would make me roll my eyes at them.) Sometimes they just cannot comprehend that they were wrong and they live in that place of denial.
The reasons are likely different person to person.
Personally – I always admit when I am wrong. Why fake it? Plus, it gives me a chance to be open with others, honest and I get to learn new things.
Reference: https://www.quora.com/How-come-you-dont-admit-when-youre-wrong