True or False questions about invertible matrices

Question

1. If $A$ is invertible, then $(A^T)^{-1}= (A^{-1})^T $

2. If $A$ and $B$ are invertible, then $A+B$ is also invertible and its inverse is $A^{-1} +B^{-1}$

Note: Given a matrix $A$, the inverse and the transpose of $A$ are denoted $A^{-1}$ and $A^T$ respectively.

Answer 1

1. If $A$ is invertible, then $ (A^T)^{-1}= (A^{-1})^T $

• true, see here Transpose of inverse vs inverse of transpose

2. If $A$ and $B$ are invertible, then $A+B$ is also invertible and its inverse is $A^{-1} +B^{-1}$

• not true, take for example $A=I$ and $B=I$

Answer 2

$1.$ $A$ is invertible $\implies$ $(A^T)^{−1}=(A^{−1})^T$

is true, because $$A (A^{-1})= I \implies (A^{-1})^TA^T =(A (A^{-1}))^T= I^T=I$$

$2.$ $A$ and $B$ are invertible $\implies A+B$ is also invertible and $(A+B)^{−1}=A^{−1}+B^{−1}$

is false, because $(A+B)^{-1}$ may not exist. For example, let $A=I$ and $B=-I$