The number of matrices A=a    bc    d  (where a,b,c,d∈R) such that A−1=−A is:

As A−1 exists,  det⁡(A)=ad−bc≠0.Also, det⁡(−A)=det⁡−a−b−c−d=ad−bc=det⁡(A)Since det⁡A−1=1det⁡(A),A−1=−Aimplies1ad−bc=ad−bc⇒(ad−bc)2=1⇒ad−bc=±1If ad−bc=1, then A−1=−AgivesA−1=d−b−ca=−a−b−c−d⇒ a+d=0∴ a(−a)−bc=1⇒ bc=−1+a2⇒bc≠0 andb=−1+a2c∴ A=c1+a2acc−a,where c≠0Thus, there are infinite number of such matrices.