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  • Let <img alt="S=\left \{ \left ( \begin{matrix} a_{11} &a_{12} \\ a_{21} & a_{22} \end{matrix} \right ) :a_{i\: j}\: \epsilon \left \{ 0,1,2 \right \},a_{11}=a_{22}\right \}" src=

Let

S=\left \{ \left ( \begin{matrix} a_{11} &a_{12} \\ a_{21} & a_{22} \end{matrix} \right ) :a_{i\: j}\: \epsilon \left \{ 0,1,2 \right \},a_{11}=a_{22}\right \}

 Then the number of  non-singular matrices in the set S s:

Option: 1

27


Option: 2

24


Option: 3

10


Option: 4

20


Answers (1)

best_answer

All the possible matrix

\begin{bmatrix} 0 &0/1/2 \\ 0/1/2& 0 \end{bmatrix}     \begin{bmatrix} 1 &0/1/2 \\ 0/1/2& 1 \end{bmatrix}    \begin{bmatrix} 2 &0/1/2 \\ 0/1/2& 2 \end{bmatrix} 

At anyplace, 0/1/2 means 0, 1 or 2 will be the element at that place. Hence there are total  27=3×3+3×3+3×3

out of which there are some singular matrix

\left[\begin{array}{cc}0 & 0 / 1 / 2 \\ 0 & 0\end{array}\right],\left[\begin{array}{cc}0 & 0 \\ 1 / 2 & 0\end{array}\right],\left[\begin{array}{cc}1 & 1 \\ 1 & 1\end{array}\right],\left[\begin{array}{cc}2 & 2 \\ 2 & 2\end{array}\right]

Here total ways are 3 + 2 + 1 + 1 = 7

So non-singular matrix will be 27 - 7 = 20

 

Posted by

Divya Prakash Singh

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