**(a) Show $J=\left(\begin{array}{rr}{0} & {-1} \\ {1} & {0}\end{array}\right) \in M_{2}(\mathbb{R}$ is a solution to $J^{2}+I=O$.**

By inserting the matrix in the matrix equation.

**(b) Are there other solutions $A \in M_{2}(\mathbb{R})$ to the above equation?**

Yes, $-J$.

**(c) Find all solutions $A \in M_{2}(\mathbb{R})$ and $B \in M_{2}(\mathbb{C})$ to $J^2+I=0$**.

Let $A \in M_{2}(\mathbb{R})$ be given by

$$A=\left(\begin{array}{rr}{a} & {b} \\ {c} & {d}\end{array}\right)$$

By inserting $A$ in the equation I get

$$\left[\begin{array}{cc}{a^{2}+b c+1} & {a b+b d} \\ {c a+d c} & {b c+d^{2}+1}\end{array}\right] = \left[\begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right]$$

** I'm stuck here**.

This means $a^{2}+b c+1=0$ etc. I get four equations in four variables. To my minds eye they look non linear. Via the Gauss-Jordan elimination method I get (calculated with Maple)

$$\left[\begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0}\end{array}\right]$$

According to Maple there are no solutions. But that does not make any sense since for example

$$J=\left(\begin{array}{cc}{0} & {-1} \\ {1} & {0}\end{array}\right)$$ is a solution.

What do I do now?