Question

Let $S=\{\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right):a_{ij}\in \{0,1,2\},a_{11}=a_{22}\}$. Then the number of non-singular matrices in the set $S$ is:

• A. $27$
• B. $24$
• C. $10$
• D. $20$

Answer 1

The correct option is D $20$

Total matrices $=3\times 3\times 3=27\left(4^{th}fixed\right)(a_{11}=a_{22})$

Checking Singular matrices (Determinant = 0)

Cases $:1)0-0\Rightarrow a_{11}=a_{22}=0;\{a_2,a_2\}c=\{0,\mathrm{0.1.2}\}$ or $\{1-2,03\}\Rightarrow \left(5\right)$ possible

$2)1-1\Rightarrow a_{11}=a_{22}=1;a_{12}=a_{21}=1\Rightarrow (1)$

$3)2-2$ (x) $a_{11}=a_{22}$ can't give 2

$4)4-4$ $a_{11}=a_{22}=2;a_{12}=a_{21}=2\Rightarrow \left(1\right)$

$\Rightarrow$ Non - singular matrices $=27-(5+1+1)$

$\Rightarrow 20\Rightarrow \left(D\right)$