is the sum of two invertible matrices necessarily invertible?

Question

is the sum of two invertible matrices necessarily invertible?
how can i prove it?

Answer 1

Absolutely not. If A is any invertible matrix, then -A is also invertible (it's easy to verify that its inverse is -(A^-1)), and A + -A = 0. The easiest example is I and -I.

Answer 2

Here's an easy way to prove that not every two invertible matrices add up to another. Take any invertible matrix A. Now, the negative of A is another invertible matrix. The sum A + (-A) is the empty matrix, which is singular. QED.

Answer 3

Scythian is absolutely correct and has given a perfect example.

An invertible matrix A, added to its negative, -A, will yield the 0 matrix, which is not invertible.

All that's needed to prove something false is a counterexample.

Answer 4

what a good question