Question

Determine the values of $a$ and $b$ for which the system $\left[\begin{array}{ccc}3& -2& 1\\ 5& -8& 9\\ 2& 1& a\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}b\\ 3\\ -1\end{array}\right]$ has a unique solution

• A. $a\ne -3;b\in R$
• B. $a\in R;b\ne -3$
• C. $a\ne 7;b\ne -3$
• D. $a\ne -5;b\in R$

Answer 1

Answer: (A)

$a\ne -3;b\in R$

$\Delta =\left[\begin{array}{ccc}3& -2& 1\\ 5& -8& 9\\ 2& 1& a\end{array}\right]=3(-8a-9)+2(5a-18)+1(5+16)=-14a-48\Delta \ne 0\Rightarrow -14a-48\ne 0\Rightarrow a\ne -3$