For what values of "a" would the system have an infinite number of solutions and which would have to satisfy b1, b2 and b3?

Question

Consider $$ \left[ \begin{array}{ccc|c} 2&-1&3&b_1\\ a&1&0&b_2\\2&1&1&b_3 \end{array} \right] $$

a) If b1, b2, b3 are not three zero, For what values of "a" would the system have an infinite number of solutions and which would have to satisfy b1, b2 and b3?

b)Suppose that b1 = b2 = b3 = 0 and determine for which values of a the system has non-trivial solutions

Hello, I will start my linear algebra class soon so I have practiced the first topics of the course to be more advanced. I suppose that in this problem I must clear and use Gauss, I have tried several times, but I can not do it. If they can help and explain how they got the result would be excellent.

Answer 1

Hint:

Put the left part of the matrix in R.R.E.F. The condition for system to have solution(s) is that the rank of this submatrix and the augmented matrix have the same rank.

When there are solution(s), this rank is the codimension of the affine subspace of solutions.

Answer 2

(a)

Must be

$ \det\left[ \begin{array}{ccc} 2&-1&3\\ a&1&0\\ 2&1&1 \end{array} \right]=0 \rightarrow 4 a-4=0 \to a=1 $

so $\mathrm{rank}A=2$

To have solutions, must be $\mathrm{rank}A|B=2$, too. So must be

$ \det\left[ \begin{array}{ccc} -1&3&b_1\\ 1&0&b_2\\1&1&b_3 \end{array} \right]=0\to b_1 + 4 b_2 - 3 b_3 = 0 $

(b)

If the system is homogeneous then it has non trivial solutions if $\det A=0$ that is for $a=1$