How to interpret $A A' = \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}$

Question

(I edited the question to be more relevant and informative/specific. Hope it's better). I'm unsure of how to interpret $$A A' = \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}$$ where A and A' are 2x2 complex singular matrices defined as $$A = \begin{pmatrix} a & b\\ c & d \end{pmatrix}$$ and $$A' = \begin{pmatrix} a & c\\ b & d \end{pmatrix}$$ Where $a, b, c, d \in \mathbb{C}$. What would this imply in terms of linear algebra. Are they orthogonal to each other? What would the nature of A be for this condition to be true?. Also, if anyone knows what field/theory these matrices are used in?

Answer 1

$$a^2+b^2=c^2+d^2=0$$ $$b=\pm ia$$ $$d=\pm ic$$ $$ac+bd=0$$ If take same sign in $\pm$, then this always hold.

Case 1: $$\left(\matrix{a&ia\\c&ic}\right)$$

Case 2: $$\left(\matrix{a&-ia\\c&-ic}\right)$$ If take opposite sign in $\pm$, then $ac=0$

Case 3: $$\left(\matrix{0&0\\c&\pm ic}\right)$$

Case 4: $$\left(\matrix{a&\pm ia\\0&0}\right)$$

Case 3 and 4 are aleady included in 1 and 2.

Hence the answer is

$$\left(\matrix{a&\pm ia\\c&\pm ic}\right)=\left(\matrix{a\\c}\right)\left(\matrix{1 & \pm i}\right)$$