sweetiepi said:Homework Statement
Let A and B be invertible matrices such that A^-1 + B^-1 is also invertible. Prove that A+B is invertible.Homework Equations
A(A^-1) = I
B(B^-1) = I
(A^-1+B^-1)(A^-1+B^-1)^-1 = I
The Attempt at a Solution
I've been trying to manipulate these equations to make something work, but I just can't seem to find the right combination.
The invertibility of a matrix sum can be proved by showing that the sum of two matrices A and B results in a matrix C, where the determinant of C is non-zero. This means that C has an inverse and thus, A+B is invertible.
Proving the invertibility of a matrix sum is important because it ensures that the sum of two matrices can be inverted, which is crucial in many mathematical and scientific calculations. It also allows for the use of various matrix operations, such as inverse and determinant, on the matrix sum.
No, a matrix sum can only be invertible if both matrices involved are invertible. If one of the matrices is not invertible, then the resulting matrix sum will also not be invertible.
The invertibility of a matrix sum is dependent on the invertibility of each individual matrix involved. If both matrices are invertible, then their sum will also be invertible. However, if one of the matrices is not invertible, then the resulting matrix sum will also not be invertible.
Yes, there are some special cases where a matrix sum may still be invertible even if one of the matrices is not invertible. For example, if one of the matrices is a zero matrix, then the matrix sum will still be invertible. Additionally, if the non-invertible matrix is a scalar multiple of the other matrix, then the matrix sum will also be invertible.