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

Question

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

Infinity Learn · Accepted Answer

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.

The number of matrices $A = [\begin{array}{ll} a b \\ c d \end{array}]$ (where $a, b, c, d \in R$ ) such that $A^{- 1} = - A$ is:

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

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The number of matrices $A = [\begin{array}{ll} a b \\ c d \end{array}]$ (where $a, b, c, d \in R$ ) such that $A^{- 1} = - A$ is: