Prove That A+B is Singular if A and B are Singular The following problem is presented: _Prove that, if A and B are singular $n \times n$ matrices, that $A+B$ is also singular._ I have the following solution, whereupon I assume that $x_1 = x_3$. ![enter image description here](

It's not true ! Take $$A=\begin{pmatrix}1&0\\\0&0\end{pmatrix}\quad \text{and}\quad B=\begin{pmatrix}0&0\\\0&1\end{pmatrix}.$$

Notice that if $A$ and $B$ are not singular, you can also have $A+B$ singular, for example $$A=\begin{pmatrix}1&0\\\0&1\end{pmatrix}\quad \text{and}\quad B=\begin{pmatrix}-1&0\\\0&-1\end{pmatrix}.$$