gauss jordan matrix involving parameter $k$

Question

Could anyone help me in solving this matrix?

$$\left[\begin{array}{ccc|c} k+2& k-1& k& 2\\ 0& k+2& 2& 0\\ 0& 0& k^2+k-2& k+2 \end{array}\right]$$

Find $k$ for which the system has:

a) exactly one solution

b) infinitely many solutions

c) no solutions

I am new in linear algebra and having trouble getting my head around this. I am particularly confused with part (b).

Answer 1

Let $$A = \begin{bmatrix} k+2 & k-1 & k\\ 0 & k+2 & 2\\ 0 & 0 & k^2 + k-2\end{bmatrix}$$ and $$[A \mid B] = \left[\begin{array}{ccc|c} k+2 & k-1 & k & 2\\ 0 & k+2 & 2 & 0\\ 0 & 0 & k^2 +k-2 & k+2\end{array}\right].$$

It is true that a system has a solution (not necessarily unique), iff: $$\boxed{\text{rank }A = \text{rank }[A\mid B]}\tag{1}.$$

• If the system has infinitely many solutions, then it must hold $\det A = 0$ and the $(1)$. We notice that $\det A = 0$ for $k = 1,-2$. Try to find the rank of $A$ and $[A\mid B]$ in each case.