How to find a matrix D such that B + D is a singular matrix.

Question

B=$\begin{pmatrix} e & 10 \\ 2 & 8 \end{pmatrix}$

How to find a matrix D such that B + D is a singular matrix.

Can anyone please show this answer

Answer 1

For $B+D$ to be singular at at least one of the rows or columns should be zero.You can also have singular matrix with non zero columns or rows but finding them requires you brute force it

So you can have $D = \begin{pmatrix}a&&-10\\b&&-8\end{pmatrix}$

$D = \begin{pmatrix}-e&&-10\\b&&a\end{pmatrix}$

$D =\begin{pmatrix}-e&&a\\-2&&b\end{pmatrix}$

$D = \begin{pmatrix} a&&b\\-2&&-8\end{pmatrix}$

$D = -B$

for any real number $a , b$