"The number of singular matrices of order 2 , whose elements are from the set $\{2,3,6,9\}$, is __________."

Question

Accepted Answer

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Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$

for $A$ to be singular matrix

$a d=b c$

Case 1: exactly 1 number is used $\Rightarrow{ }^4 C_1$ ways

Case 2 : exactly 2 numbers is used $\Rightarrow{ }^4 C_2$ ways

Case 3 : exactly 3 numbers used $\Rightarrow$ none will be singular.

Case 4: exactly 4 numbers is used

$$\begin{aligned} & \Rightarrow a b=c d \Rightarrow 2 \times 9=3 \times 6 \\ & \Rightarrow{ }^4 C_1 \times 2!=8 \text { matrix } . \end{aligned}$$

$\therefore$ Total ways $\Rightarrow 4+6 \times 4+8=36$ matrices.

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The number of singular matrices of order 2 , whose elements are from the set $\{2,3,6,9\}$, is __________.