The system of linear equations x+y+z=2,      2x+3y+2z=5       2x+3y+a2−1z=a+1

According to cramer’s rule , here D=11123223a2−1=10021021a2−3 ( applying C2→C2−C1 and C3→C3−C1=a2−3         ( Expanding along  R1)  And  D1=211532a+13a2−1=21053−1a+13a2−1−3 ( Applving C2→C2−C2=20053−52−1a+13−a+12a2−1−3 ( Applving C2→C2−C2=200512−1a+152−a2a2−4=212a2−4+52−a2 expanding along R1=2a22−2+52−a2=a2−4+5−a=a2−a+1 Clearly, when a=4, then D=13≠0⇒ unique solution and  When |a|=3, then D=0  and D1≠0∴ when |a|=3, then the system has no solution i.e. system is inconsistent.