Mathematics GATE 2018 EC: 22

Question

Consider matrix $A =$ $\begin{bmatrix} k &2k \\ k^{2}-k &k^{2} \end{bmatrix}$ and $x =$ $\begin{bmatrix} x1\\x2 \end{bmatrix}$

The number of distinct real values of $k$  for which the equation  $Ax = 0$
has infinitely many solutions is _______.

Answer 1

Answer should be 2

System of Linear Homogeneous Equations ie AX = 0 has infinite number of solutions when |A| = 0 ie. rank of the coefficient matrix is less than the no. of unknown variables. 

So ,  Here , |A| = 0 

So , After Expanding , k*k² - (k² - k)*(2k) = 0

k³ - 2k³ + 2k² = 0

-k³ + 2k² = 0

k³ - 2k²  = 0

k²(k-2) = 0 . So distinct values of k = 0,2