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

Step 1: Define a singular matrix

A square matrix $A$ is singular if its determinant is equal to zero. For a $2 \times 2$ matrix, the determinant is calculated as $ad - bc$. So, we are looking for matrices where $ad - bc = 0$, which means $ad = bc$. The elements $a, b, c, d$ must come from the set $\{2, 3, 6, 9\}$.

Step 2: List possible products

We need to find pairs of elements $(a, d)$ and $(b, c)$ from the set $\{2, 3, 6, 9\}$ such that their products are equal. Let's list all possible products of two elements from this set. The set of elements is $S = \{2, 3, 6, 9\}$.

Step 3: Calculate all possible products

Let's create a multiplication table for the elements in the set $\{2, 3, 6, 9\}$ to find all possible products. This table shows all $4 \times 4 = 16$ possible products when choosing two elements from the set, with replacement and order matters for the elements in the product.

Step 4: Identify equal products and count matrices

Now we need to find pairs of products $(ad, bc)$ that are equal. We will count the number of ways to choose $a, d, b, c$ such that $ad=bc$. We can categorize these cases based on the value of the product: 1. Product = 4: Only $2 \times 2 = 4$. So $ad=4$ has 1 way ($(a,d)=(2{,}2)$) and $bc=4$ has 1 way ($(b,c)=(2{,}2)$). This gives $1 \times 1 = 1$ matrix. 2. Product = 6: $2 \times 3 = 6$ and $3 \times 2 = 6$. So $ad=6$ has 2 ways, and $bc=6$ has 2 ways. This gives $2 \times 2 = 4$ matrices. 3. Product = 9: $3 \times 3 = 9$. So $ad=9$ has 1 way, and $bc=9$ has 1 way. This gives $1 \times 1 = 1$ matrix. 4. Product = 12: $2 \times 6 = 12$ and $6 \times 2 = 12$. So $ad=12$ has 2 ways, and $bc=12$ has 2 ways. This gives $2 \times 2 = 4$ matrices. 5. Product = 18: $2 \times 9 = 18$, $3 \times 6 = 18$, $6 \times 3 = 18$, $9 \times 2 = 18$. So $ad=18$ has 4 ways, and $bc=18$ has 4 ways. This gives $4 \times 4 = 16$ matrices. 6. Product = 27: $3 \times 9 = 27$ and $9 \times 3 = 27$. So $ad=27$ has 2 ways, and $bc=27$ has 2 ways. This gives $2 \times 2 = 4$ matrices. 7. Product = 36: $6 \times 6 = 36$. So $ad=36$ has 1 way, and $bc=36$ has 1 way. This gives $1 \times 1 = 1$ matrix. 8. Product = 54: $6 \times 9 = 54$ and $9 \times 6 = 54$. So $ad=54$ has 2 ways, and $bc=54$ has 2 ways. This gives $2 \times 2 = 4$ matrices. 9. Product = 81: $9 \times 9 = 81$. So $ad=81$ has 1 way, and $bc=81$ has 1 way. This gives $1 \times 1 = 1$ matrix. Summing these up: $1 + 4 + 1 + 4 + 16 + 4 + 1 + 4 + 1 = 36$.

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