Q.If the matrix $A$ is both symmetric and skew symmetric, then (A) $A$ is a diagonal matrix (B) $A$ is a zero matrix (C) $A$ is a square matrix (D) None of these

The key idea is that a matrix that is both symmetric ($A^T = A$) and skew-symmetric ($A^T = -A$) forces every element to equal its own negative.

1. From the two conditions, equate the right-hand sides: $A = -A$.
2. Adding $A$ to both sides gives $2A = 0$, so $A = 0$ (the zero matrix). …

The key here is to understand what it means for a matrix to be symmetric and skew-symmetric at the same time. These are two different properties that usually describe opposite kinds of matrices — one is equal to its transpose, the other is equal to the negative of its transpose. When both hold simultaneously, the only possible matrix is the zero matrix.

Let’s break it down step by step.

1. Recall the definitions.

   A square matrix $A$ is symmetric if $A = A^T$. That means $a_{ij} = a_{ji}$ for all $i, j$.

   A square matrix $A$ is skew-symmetric if $A = -A^T$. That means $a_{ij} = -a_{ji}$ for all $i, j$.

2. Apply both conditions together.

   If $A$ is both symmetric and skew-symmetric, then for every entry $a_{ij}$ we have:

and also

3. Combine the two equations. From the two equalities, we get:

Adding $a_{ij}$ to both sides gives:

This holds for every $i$ and $j$, meaning every entry of $A$ is zero.

4. What about diagonal entries? For $i = j$, the skew-symmetric condition says $a_{ii} = -a_{ii}$, which also forces $a_{ii} = 0$. So even the diagonal is all zeros. …

Method: Combining two defining conditions on a matrix

Use this reasoning pattern when a matrix is required to satisfy two structural definitions at once (here: symmetric and skew-symmetric).

Steps

Step 1: Write each definition as an equation on the entries.

Symmetric means $a_{ij} = a_{ji}$; skew-symmetric means $a_{ij} = -a_{ji}$.

Step 2: Combine the two equations.

Substituting one into the other gives $a_{ij} = -a_{ij}$, hence $2a_{ij} = 0$ for every $i,j$. …

Common Mistakes

Mistake 1: Answering "diagonal matrix".

Why it's wrong: skew-symmetry forces the diagonal to be zero and symmetry+skew-symmetry forces every entry to zero — a general diagonal matrix has non-zero diagonal entries. Correct approach: conclude it is the zero matrix.

Mistake 2: Choosing the broader-but-weaker "square matrix".

Why it's wrong: being square is necessary for the definitions to apply but does not capture what the matrix must be. Correct approach: pick the strongest correct statement — the zero matrix (which is, of course, also square). …