Prove That A+B is Singular if A and B are Singular

Question

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 $x_1 = x_3$.

Answer 1

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 $$A=\begin{pmatrix}1&0\\0&1\end{pmatrix}\quad \text{and}\quad B=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}.$$

Answer 2

You cannot prove this as it is false. A counter example is given in another answer. To complement it let me point to the specific error you make: It is when you say $Ax_3=0$. There is no reason that $Ax_3= 0$. There is some $x_2$ such that $Ax_2=0$ but this might well be different from $x_3$, and you cannot chose the $x$ as you want. The are determined by $A$ and $B$.