The system of linear equaltions.

$x+y+z=0$

$2x+3y+2z=5$

$2x+3y+(a^2-1)z=a+1$

(1) has infinitely many solutions for $a=4$

(2) is inconsistent when $|a|=\sqrt3$

(3) is inconsistent when $a=4$

(4) has a unique solution for $|a|=\sqrt3$

$x+y+z=0$

$2x+3y+2z=5$

$2x+3y+(a^2-1)z=a+1$

(1) has infinitely many solutions for $a=4$

(2) is inconsistent when $|a|=\sqrt3$

(3) is inconsistent when $a=4$

(4) has a unique solution for $|a|=\sqrt3$