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

Q: Why do we split this into cases based on distinct elements? Can't we just write a code or list them all?

Listing 256 matrices is a recipe for disaster and prone to human error. By partitioning the problem into cases based on the number of distinct elements, we impose a logical structure that makes the counting systematic and elegant. It transforms a brute-force nightmare into a series of manageable, bite-sized combinatorial puzzles.

Q: In Case 3, I thought there might be a solution. Why is it zero?

It is a classic trap! When you use three distinct elements, one must be repeated. If we let $x$ be the repeated element, the condition $ad = bc$ becomes $x^2 = yz$, where $y$ and $z$ are the other two distinct elements. If you carefully test all combinations from the set $\{2, 3, 6, 9\}$, you will find that no square of one element equals the product of two others. It is a beautiful example of how a seemingly possible case can be mathematically impossible.

Q: How do I know I haven't missed any arrangements in Case 4?

Think about the symmetry of the determinant. We found the pairs $\{2, 9\}$ and $\{3, 6\}$ because $2 \times 9 = 18$ and $3 \times 6 = 18$. For the main diagonal $\{a, d\}$, we can choose either $\{2, 9\}$ or $\{3, 6\}$. If we choose $\{2, 9\}$, the off-diagonal $\{b, c\}$ must be $\{3, 6\}$. Within each pair, there are $2!$ ways to arrange them, and there are $2$ ways to swap the diagonals themselves. This gives $2 \times 2 \times 2 = 8$ total arrangements, covering all possibilities.

Question

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

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a $2 	imes 2$ matrix, a simple grid of four numbers. You are tasked with filling these slots using only the numbers from the set $S = \{2, 3, 6, 9\}$.

The matrix must be singular, which means its determinant is zero. For a matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is given by $|A| = ad - bc$.

For the matrix to be singular, we must satisfy the condition $ad - bc = 0$, or more simply:

$$ad = bc$$

This is the heartbeat of our problem. We are not just filling slots; we are hunting for pairs of products that match.

### The Strategy of Cases

Trying to check all $4^4 = 256$ possible matrices would be a tedious, error-prone endeavor. Instead, let us be strategic.

We will categorize our search based on the number of distinct elements used. This is the mark of a true mathematician: breaking a complex problem into smaller, solvable pieces.

### Case 1: The Uniformity

What if all four elements are identical? Let $a = b = c = d = x$.

The condition $ad = bc$ becomes $x^2 = x^2$, which is always true. Since we have four choices for $x$ from our set $\{2, 3, 6, 9\}$, we immediately find **4** singular matrices.

### Case 2: The Two-Element Dance

Now, suppose we use exactly two distinct elements, $x$ and $y$. We need $ad = bc$ using only these two.

The only way to satisfy this is if the products are equal, which happens when $ad = bc = xy$. This leads to four specific arrangements for any pair $\{x, y\}$:

$$\begin{bmatrix} x & x \ y & y \end{bmatrix}, \begin{bmatrix} y & y \ x & x \end{bmatrix}, \begin{bmatrix} x & y \ x & y \end{bmatrix}, 	ext{ and } \begin{bmatrix} y & x \ y & x \end{bmatrix}$$

Since there are $\binom{4}{2} = 6$ ways to choose two distinct elements from our set, and each pair gives us 4 matrices, we have $6 	imes 4 =$ **24** singular matrices here.

### Case 3: The Empty Trap

What about using three distinct elements? One must be repeated. Let $x$ be the repeated element.

The condition $ad = bc$ becomes $x^2 = yz$. We must check if the square of any element in our set equals the product of two other distinct elements.

Testing $2^2=4$, $3^2=9$, $6^2=36$, and $9^2=81$ against the products of the remaining elements reveals no matches. This case is a ghost; it yields **0** matrices.

### Case 4: The Perfect Match

Finally, we consider the case where all four elements are distinct. We must use all four numbers: $2, 3, 6, 9$.

We need to partition them into two pairs with equal products. Looking at the numbers, we see $2 	imes 9 = 18$ and $3 	imes 6 = 18$.

We can place the pair $\{2, 9\}$ on the main diagonal in $2!$ ways, and the pair $\{3, 6\}$ on the off-diagonal in $2!$ ways, giving $2 	imes 2 = 4$ matrices. We can also swap the diagonals, placing $\{3, 6\}$ on the main diagonal and $\{2, 9\}$ on the off-diagonal, giving another 4 matrices.

Total for this case: $4 + 4 =$ **8**.

### The Grand Finale

We have traversed the landscape of possibilities. Adding our results:

$$4 	ext{ (Case 1)} + 24 	ext{ (Case 2)} + 0 	ext{ (Case 3)} + 8 	ext{ (Case 4)} = 36$$

There are exactly **36** singular matrices. It is a beautiful result, born not from brute force, but from the structured, logical beauty of case analysis.

Animated Solution for Mathematics - Matrices and Determinants: The number of singular matrices of order 2, whose elements are from the set ${2, 3, 6, 9}$ is _____

Enter Numerical Value:

Visualized Solution

The Matrix and the Set

Condition for a Singular Matrix

Expanding the Determinant

Strategy: Case Analysis

Case 1: One Distinct Element

Case 1: Total Matrices

Case 2: Two Distinct Elements

Case 2: Possible Structures

Case 2: Total Matrices

Case 3: Three Distinct Elements

Case 3: Checking $x^{2} = y z$

Case 4: Four Distinct Elements

Case 4: Matching Products

Case 4: Arranging the Elements

Total Number of Singular Matrices

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Strategy of Cases

Case 1

Case 2

Case 3

Case 4

The Grand Finale

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Animated Solution for Mathematics - Matrices and Determinants: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of singular matrices of order 2, whose elements are from the set {2,3,6,9} is _____