The matrix B^2 is:



To make B^2 = I3, the following must be true:

2b21 = 0 --> b21 = 0

2b32 = 0 --> b32 = 0

2b31 + b21b32 = 0 --> 2b31 + 0*0 = 0 --> b31 = 0

Thus, b21, b31, and b32 must all be 0.

Edit in response to follow-up comment from OP:

If b22 = -1 instead of b22 = 1 as originally posted, then B^2 would be:



In this case, b21 and b32 could be any value to make B^2 = I3. In addition,
b31 = -(b21 * b32)/2.

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