Matrices (A+B)^-1 = A^-1+B^-1 is not true for all square matrices A,B

Hi,

I have this question and I am not sure how to tackle it:

Construct an infinite number of examples that demonstrate (A+B)^-1 = A^-1 + B^-1 is not true for all square matrices A and B of the same size.

So I am aware from the Linear Algebra unit we have been doing that if A or B are not invertible then their inverse does not exist but I am unsure how to use this to construct an infinite number of examples. Any help would be greatly appreciated! Thanks in advance

Hey Cotty.

One example is setup two matrices: the first has a full row of zeros in the first row and the second has a full row of zeroes in the second row. Together though, the sum of the matrices gives a matrix that is invertible, however each matrix on its own is not.

I think the question (from the way it’s written) assumes that all of , and are invertible.

Hint: Let and be multiples of the identity matrix.

You're amazing I got it, thank you so much!!!!! Thanks Chiro as well