Is the sum of two singular matrices also singular? If $A$ and $B$ are $n \times n$ singular matrices, is $A+B$ also singular?


The sum of two singular $n × n$ matrices may be non-singular.

For example, consider two $2 × 2$ matrix $A=\begin{pmatrix} 
1 & 1 \\
1 & 1 
\end{pmatrix}$ and $B=\begin{pmatrix} 
1 & -1 \\
-1 & 1 
\end{pmatrix}$
Here $\det A = 1-1=0$ and $\det B=1-1=0$
So both the matrix $A \quad \text{and} \quad B$ are singular matrices. 
Now $A+B =\begin{pmatrix} 
2 & 0 \\
0 & 2 
\end{pmatrix}$
$\det (A+B)=4\neq 0$
Hence $A + B$ is non-singular. 
