Question

If 4 elements are randomly selected from the collection $\{-2,0,1,2,0\}$ and $2 \times 2$ matrices are formed using each of the selected elements, then the chance that the selected $2 \times 2$ matrix is invertible, is

• A. $\frac{2}{5}$
• B. $\frac{7}{15}$
• C. $\frac{1}{5}$
• D. $\frac{3}{5}$

Answer 1

Answer: (D)

4 randomly selected elements can be $-2,0,2,0$ ( 1 is not taken) $\Rightarrow 4$ are invertible $-2,0,1,0 \quad(2$ is not taken) $\Rightarrow 4$ are invertible $0,1,2,0 \quad(-2$ is not taken $) \Rightarrow 4$ are invertible $-2,0,1,2$ (one 0 's is not taken) $\Rightarrow 24$ are invertible Total $(2 \times 2$ matrices $)=3\left(\frac{4 !}{2 !}\right)+4$ !
$\therefore \quad n ( S )=36+24=60$
$\therefore P(A)=\frac{36}{60}=\frac{3}{5}$