find all values of a such that the system Axb has a unique s

find all values of \"a\" such that the system Ax=b has a unique solution for any b in element R^3 ( must use determinant).

Solution

given A=

3a.....0.......0

a.......4......2a

1.......2a......1

Ax=b has a unique solution for any b when determinant of A0

detA=3a[(4*1)-(2a*2a)]+0[(1*2a)-(a*1)]+0[(a*2a)-(1*4)]0

3a[4-4a²]+0+00

12a[1-a²]0

a0,a-1,a1

so system has unique solution when a belong to (-,-1)U(-1,0)U(0,1)U(1,)