Is the sum of two singular matrices also singular?

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If $A$ and $B$ are $n \times n$ singular matrices, is $A+B$ also singular?

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2 Answers

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No. Split up the identity matrix.

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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.

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