Question

Consider for every real number a the linear system of equations:


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



b)Find the values of a for which the system has no solution, infinitely many solutions, and a unique solution

Answer 1

Judging from what wolframalpha gets at

http://www.wolframalpha.com/input/?i=solve+x+%2B%28+a+%2B+1+%29y+%2B+a^2+z+%3D+a^3%2C+%28+1+-+a+%29x+%2B%28+1+-+2a+%29y+%3D+a^3%2C+x+%2B%28+a+%2B+1+%29y+%2B+az+%3D+a^2

I suspect a typo.

Answer 2

note first that if a = 1, the first and third rows are identical (infinite number of solutions)

then
calculate the determinant of the coefficient matrix. Find its zeros.

One will be a = 0 Look at second equation when a = 0, x+y = 0 and first equation gives x+y = 0 and third equation gives x+y = 0 so any old x = -y will do for any old z