Let S = ( * 20 c a_ 11 & a_ 12 a_ 21 & a_ 22 ): a_ ij 0,1,2 , a_ … - Vidyadip

MCQ

Let $S = \left\{ {\left( {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}\\
{{a_{21}}}&{{a_{22}}}
\end{array}} \right):{a_{ij}} \in \left\{ {0,1,2} \right\},{a_{11}} = {a_{22}}} \right\}$ Then the number of non-singular matrices in the set $S$ is

A
$27$
B
$24$
C
$10$
✓
$20$

✓

Answer

Correct option: D.

$20$

d
The matrices in the form

$\left[ {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}\\
{{a_{21}}}&{{a_{22}}}
\end{array}} \right],{a_{ij}} \in \left\{ {0,1,2} \right\},{a_{ 11}} = {a_{12}}$ are

$\left[ {\begin{array}{*{20}{c}}
0&{0/1/2}\\
{0/1/2}&0
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
1&{0/1/2}\\
{0/1/2}&1
\end{array}} \right].\left[ {\begin{array}{*{20}{c}}
2&{0/1/2}\\
{0/1/2}&2
\end{array}} \right]$

At any place, $0/1/2$ means $0,1$ or $2$ will be the element at that place.

Hence therefore total $\left( {27 = 3 \times 3 + 3 \times 3 + 3 \times 3} \right)$

matrices of the above form. Out of which the matrices which are singular are

$\left[ {\begin{array}{*{20}{c}}
0&{0/1/2}\\
0&0
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
0&0\\
{1/2}&0
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
1&1\\
1&1
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
2&2\\
2&2
\end{array}} \right]$

Hence these are total $7=(3+2+1+1)$ singular matrices.

Therefore number of all non-singular matrices in the given form $=27-7=20$

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