The system of linear equations
x + y + z = 2
2x + 3y + 2z = 5
`2x + 3y + (a^(2) - 1)z = a + 1`

Options

A. has infinitely many solutions for a = 4
B. is inconsistent when a=4
C. has a unique solution for `|a| =sqrt(3)`
D. is inconsistant when `|a| = sqrt(3)`

Explanation

According to Cramer's rule, here
`D = |{:(1, 1, " "1), (2, 3, " "2), (2, 3, a^(2)-1):}| = |{:(1, 0, " "0), (2, 1, " "0), (2, 1, a^(2)-3):}|`
`("Applying" C_(2) to C_(2)-C_(1) " and " C_(3) to C_(3)-C_(1))`
`= a^(3) -3 " " ("Expanding along "R_(1))`
`"and " D_(1) = |{:(2, 1, " "1), (5, 3, " "2), (a+1, 3, a^(2)-1):}| = |{:(2, 1, " "0), (5, 3, " "-1), (a+1, 3, a^(2)-1-3):}|`
`("Applying"C_(3) to C_(8) - C_(2))`
` = |{:(2, " "0, " "0), (5, 3-(5)/(2), " "-1), (a+1, 3-((a+1))/(2), a^(2)-1-3):}|`
`("Applying "C_(2) to C_(2) - (1)/(2)C_(1))`
` = |{:(2, " "0, 0), (5, " "(1)/(2), -1), (a+1, (5)/(2)-(a)/(2), a^(2)-4):}|`
` = 2[(1)/(2)(a^(2) -4) + ((5)/(2)-(a)/(2))] " "["Expanding along"R_(1)]`
` = 2[(a^(2))/(2)-2 + (5)/(2) -(a)/(2)] =a^(2) -4 +5 -a =a^(2) -a + 1`
Clearly, when a =4, then `D= 13 ne 0 rArr` unique solution and
When `|a| = sqrt(3), " then"D = 0 " and "D_(1) ne 0.`
`therefore " When " |a| = sqrt(3)`, then the system has no solution i.e. system is inconsistent.