Question

Consider matrix $A=\left[\begin{array}{ccc}k& 2k& \\ k^2-k& k^2& \end{array}\right]$and vector x=$\left[\begin{array}{cc}{x}_1& \\ x_2& \end{array}\right]$. The number of distinct real values of k for which the equation Ax=0 has infinitely many solutions is ______ .

• A. 2

Answer 1

The correct option is A 2
$A=\left[\begin{array}{cc}k& 2k\\ k^2-k& k^2\end{array}\right]$ and given system is AX = 0
We know that homogeneous system have infinite solution if |A|=0
$k^3-2k(k^2-k)=0$
$\Rightarrow k^3-2k^3+2k^2=0$
$\Rightarrow k^2(2-k)=0$
$\Rightarrow k=0,0,2$
Hence k have two distinct values.