Math Help - help needed!!! proof of invertible matrices

The question has been attached as an image.

any help or light on this question would be hugely appreciated!!

thank you

Originally Posted by kaya2345

The question has been attached as an image.
any help or light on this question would be hugely appreciated!!
thank you

If and .

Is invertible?

Can we take away any of those ones?

Originally Posted by Plato

If and .

Is invertible?

Can we take away any of those ones?

thank you. and yes In is invertible

That shows, I think, that the minimum number of 1s in such an invertible n by n matrix is n. But the question was about the maximum number. We do know that if a matrix has two columns the same, it is not invertible. I think we can avoid that by having a 0, at a different row, in each column. That implies that the maximum number of 1s in such an invertible n by n matrix is .

Originally Posted by HallsofIvy

That shows, I think, that the minimum number of 1s in such an invertible n by n matrix is n. But the question was about the maximum number. We do know that if a matrix has two columns the same, it is not invertible. I think we can avoid that by having a 0, at a different row, in each column. That implies that the maximum number of 1s in such an invertible n by n matrix is .

Thank you, for that. Could I prove this using proof by induction? or contradiction?

Originally Posted by HallsofIvy

That shows, I think, that the minimum number of 1s in such an invertible n by n matrix is n. But the question was about the maximum number.

That mistake was once caused by my laptop not doing well with those attachments.
I wish we could outlaw the posting of questions in images.