Det[IdentityMatrix[4]-(({{0, -1+X, -1+X, 0}, {0, 0, -1+X, 0}, {-1+X, 0, 0, -1+X}, {-1+X, 0, 0, 0}}))]

allsigns=Det[IdentityMatrix[4]-(({{0, #[[1]], #[[2]], 0}, {0, 0, #[[3]], 0}, {#[[4]], 0, 0, #[[5]]}, {#[[6]], 0, 0, 0}}))(X-1)]& /@ ((2IntegerDigits[#,2,8]-1)& /@ Range[0,63])

allsigns /. X->2