Q._________ matrix is both symmetric and skew symmetric matrix.

The key idea is that a matrix which is both symmetric ($A^T = A$) and skew-symmetric ($A^T = -A$) must satisfy $A = -A$, which forces every entry to be zero.

Step 1: For a symmetric matrix, $a_{ij} = a_{ji}$.

Step 2: For a skew-symmetric matrix, $a_{ij} = -a_{ji}$. …

Why This Question Matters

This is a classic "boundary case" in matrix theory. Most matrices are either symmetric ($A^T = A$) or skew-symmetric ($A^T = -A$), but very few can be both. The question tests whether you understand that these two conditions together force every single entry to be zero.

Let’s see why.

Step-by-Step Reasoning

1. Write down what each condition means for an entry $a_{ij}$.

Let $A = [a_{ij}]$ be an $n \times n$ matrix.

• Symmetric: $A^T = A$ means $a_{ji} = a_{ij}$ for all $i, j$.
• Skew-symmetric: $A^T = -A$ means $a_{ji} = -a_{ij}$ for all $i, j$.

2. Combine the two conditions.

If $A$ is both symmetric and skew-symmetric, then for every pair $(i, j)$:

Equating the right-hand sides:

3. Solve for $a_{ij}$.

Add $a_{ij}$ to both sides:

This holds for every $i$ and $j$. So every entry of $A$ is zero. …

Method: Combine two definitions to force entry values

Use this reasoning pattern when a matrix must satisfy two definitions at once (here, symmetric and skew-symmetric).

Steps

Step 1: Write down each definition at entry level.

Symmetric: $a_{ij} = a_{ji}$. Skew-symmetric: $a_{ij} = -a_{ji}$.

Step 2: Set the two conditions equal and solve for the entries.

If both hold, then $a_{ji} = a_{ij}$ and $a_{ji} = -a_{ij}$ give …

Common Mistakes

Mistake 1: Answering "identity" or "unit" matrix.

Why it's wrong: the identity is symmetric but not skew-symmetric (its diagonal is $1$, not $0$). Correct approach: the only matrix that is both is the zero matrix.

Mistake 2: Forgetting the diagonal entries. …