Give an example of a skew–symmetric2×2–matrix B with entries in C for which I2+B is not invertible

Question

Give an example of a skew-symmetric $2\times2$ matrix $B$ with entries in $\mathbb C$ for which $I_2+B$ is not invertible.

I'm struggling with this Lin Algebra problem if you could help me with it that'd be great. Thank you.

Answer 1

Hints: (1) A skew symmetric $2\times 2$ matrix has entries $a,b,-b,a$

(2) $I_2$ has entries $1,0,0,1$

(3) A square matrix is not invertible if and only if its determinant is zero

Answer 2

You need to put two concepts together: 1. what is a skew symmetric matrix? and 2. When is a matrix not invertible? The answer for 1 is a matrix of type $$B=\begin{pmatrix}0 &b\\ -b&0\end{pmatrix}$$ The answer for 2 is $$\det(I_2+B)=0$$ Calculate explicitly this determinant, and see when it is $0$.