Collection of Maths Problems

System with a parameter

Task number: 2389

• Hint

  Use the eliminatin, but avoid division by zero. This might appear for some choices of $a$.

• Resolution

  We first switch the first and the last rows, and then reduce the first column. In the next reduction we add the second and the third row to the last one.

  $\left(\begin{array}{cccc|c} a&1&1&1&1\\ 1 & a & 1 & 1 & 1\\ 1 & 1 & a & 1 & 1\\ 1 & 1 & 1 & a & 1\\ \end{array}\right)\sim \left(\begin{array}{cccc|c} 1 & 1 & 1 & a & 1\\ 0 & a-1 & 0 & 1-a & 0\\ 0 & 0 & a-1 & 1-a & 0\\ 0 & 1-a & 1-a & 1-a^2 & 1-a\\ \end{array}\right)\sim\\ \left(\begin{array}{cccc|c} 1 & a & 1 & 1 & 1\\ 0 & a-1 & 0 & 1-a & 0\\ 0 & 0 & a-1 & 1-a & 0\\ 0 & 0 & 0 & (a+3)(1-a) & 1-a\\ \end{array}\right)$

  Now it suffices to ditinguish cases $a=-3$ and $a=1$.

• Result

  When $a=1$ the system has solution set $\{(1-p-q-r,p,q,r)^T\mid p,q,r\in\mathbb R\}$.

  For $a=-3$ the system has no solution.

  Otherwise the solution is $(\frac{1}{a+3},\frac{1}{a+3},\frac{1}{a+3},\frac{1}{a+3})^T$.