Let M denote the set of all real matrices of order 3x3 and let S = {-3, -2, -1, 1, 2}. Let
S_1 = {A=

Q: Wait, why is $n(S_2) = 0$? Can't we just pick other numbers for the diagonal?

That is a fantastic question! In a skew-symmetric matrix, the definition $A = -A^T$ forces the diagonal elements to satisfy $a_{ii} = -a_{ii}$, which implies $2a_{ii} = 0$, or $a_{ii} = 0$. Since $0$ is not in our set $S = \{-3, -2, -1, 1, 2\}$, it is physically impossible to construct even a single skew-symmetric matrix. The set $S_2$ is truly empty.

Q: How did you get 12 ways for the diagonal in $S_3$? I keep missing combinations.

Think of it as a partition problem. We need three numbers from $S$ that sum to zero. We have three cases: 1) Three distinct numbers: $\{1, 2, -3\}$ (sum is 0). These can be arranged in $3! = 6$ ways. 2) Two identical numbers: $\{1, 1, -2\}$ (sum is 0). These can be arranged in $3!/2! = 3$ ways. 3) Another pair of identical numbers: $\{-1, -1, 2\}$ (sum is 0). These also give $3!/2! = 3$ ways. Total = $6 + 3 + 3 = 12$.

Q: Why is the intersection $n(S_1 \cap S_3) = 12 \times 5^3$?

Great intuition! $S_1 \cap S_3$ represents matrices that are both symmetric AND have a trace of zero. We already know there are 12 ways to set the diagonal to satisfy the trace condition. Because the matrix must be symmetric, once you choose the three upper-triangular elements ($a_{12}, a_{13}, a_{23}$), the lower-triangular elements are fixed by symmetry. Each of those three positions has 5 choices, hence $5^3$.

Question

Let M denote the set of all real matrices of order $3	imes3$ and let $S=\{-3,-2,-1,1,2\}$. Let
$S_{1}=\{A=[a_{ij}]\in M:A=A^{T} 	ext{ and } a_{ij}\in S,\forall i,j\}$
$S_{2}=\{A=[a_{ij}]\in M:A=-A^{T} 	ext{ and } a_{ij}\in S,\forall i,j\}$
$S_{3}=\{A=[a_{ij}]\in M:a_{11}+a_{22}+a_{33}=0 	ext{ and } a_{ij}\in S,\forall i,j\}$
If $n(S_{1}\cup S_{2}\cup S_{3})=125\alpha$ then $\alpha$ equals

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are working with the set $M$ of all $3 	imes 3$ matrices where each entry is chosen from the set $S = \{-3, -2, -1, 1, 2\}$. The total number of possible matrices is $5^9$.

We define three subsets:
*   $S_1$: The set of symmetric matrices ($A = A^T$).
*   $S_2$: The set of skew-symmetric matrices ($A = -A^T$).
*   $S_3$: The set of trace-zero matrices ($	ext{tr}(A) = 0$).

Our goal is to find the size of the union $n(S_1 \cup S_2 \cup S_3)$.

### The Symmetric Mirror ($S_1$)

A symmetric matrix is defined by $a_{ij} = a_{ji}$. The diagonal elements ($a_{11}, a_{22}, a_{33}$) and the upper triangular elements ($a_{12}, a_{13}, a_{23}$) can be chosen independently.

Once these 6 positions are filled, the lower triangular elements are fixed. Since each of the 6 positions has 5 choices:

$$n(S_1) = 5^6 = 15625$$

### The Empty Set ($S_2$)

For a skew-symmetric matrix, the condition $A = -A^T$ implies $a_{ii} = -a_{ii}$, which forces all diagonal elements to be $0$.

However, the set $S = \{-3, -2, -1, 1, 2\}$ does not contain $0$. Therefore, it is impossible to form a skew-symmetric matrix under these constraints.

$$n(S_2) = 0$$

### The Trace-Zero Constraint ($S_3$)

For $S_3$, we require $a_{11} + a_{22} + a_{33} = 0$. We identify the valid combinations of diagonal elements from $S$:

1.  **Distinct elements:** $\{1, 2, -3\}$ sums to $0$. These can be arranged in $3! = 6$ ways.
2.  **Two identical elements:** $\{1, 1, -2\}$ sums to $0$. These can be arranged in $\frac{3!}{2!} = 3$ ways.
3.  **Another pair of identical elements:** $\{-1, -1, 2\}$ sums to $0$. These can be arranged in $\frac{3!}{2!} = 3$ ways.

The total number of valid diagonal configurations is $6 + 3 + 3 = 12$. The remaining 6 entries in the matrix can be any of the 5 values in $S$:

$$n(S_3) = 12 	imes 5^6$$

### The Intersection and the Grand Finale

Using the Principle of Inclusion-Exclusion, and noting that $n(S_2) = 0$:

$$n(S_1 \cup S_3) = n(S_1) + n(S_3) - n(S_1 \cap S_3)$$

To find $n(S_1 \cap S_3)$, we consider symmetric matrices with trace zero. We have 12 ways to choose the diagonal. For the remaining 6 non-diagonal entries, symmetry dictates that we only have 3 independent choices (the upper triangle), each with 5 options:

$$n(S_1 \cap S_3) = 12 	imes 5^3$$

Now, we assemble the final expression:

$$n(S_1 \cup S_3) = 5^6 + 12 	imes 5^6 - 12 	imes 5^3$$

Factoring out $5^3$:

$$n(S_1 \cup S_3) = 5^3 (5^3 + 12 	imes 5^3 - 12)$$
$$n(S_1 \cup S_3) = 125 (125 + 1500 - 12)$$
$$n(S_1 \cup S_3) = 125 	imes 1613$$

Given the form $125\alpha$, we conclude that **$\alpha = 1613$**.

Enter Numerical Value:

Visualized Solution

Understanding Set $S$ and Matrix Sets

Analyzing Set $S_{1}$ : Symmetric Matrices

Analyzing Set $S_{2}$ : Skew-Symmetric Matrices

Analyzing Set $S_{3}$ : Trace Zero Matrices

Finding Diagonal Combinations

Calculating $n (S_{3})$

Intersection $n (S_{1} \cap S_{3})$

Principle of Inclusion-Exclusion

Final Calculation for $α$

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Symmetric Mirror ( $S_{1}$ )

The Empty Set ( $S_{2}$ )

The Trace-Zero Constraint ( $S_{3}$ )

The Intersection and the Grand Finale

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Enter Numerical Value:

Visualized Solution

Understanding Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S and Matrix Sets

Analyzing Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S1​: Symmetric Matrices

Analyzing Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S2​: Skew-Symmetric Matrices

Analyzing Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S3​: Trace Zero Matrices

Finding Diagonal Combinations

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(S3​)

Intersection .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(S1​∩S3​)

Principle of Inclusion-Exclusion

Final Calculation for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α

The Sigma Insight: Types of Matrices

Analyzing the Setup

The Symmetric Mirror ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S1​)

The Empty Set ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S2​)

The Trace-Zero Constraint ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S3​)

The Intersection and the Grand Finale

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