Q.If $A$ is skew symmetric, then $kA$ is a _________. ($k$ is any scalar)

The key idea is that a skew-symmetric matrix satisfies $A^T = -A$. Multiplying by a scalar $k$ preserves this property.

1. For a skew-symmetric matrix $A$, we have $A^T = -A$.
2. Consider $(kA)^T$. Using the property $(kA)^T = k A^T$, we substitute $A^T = -A$: $(kA)^T = k(-A) = -(kA).$ …

The key here is to understand what "skew-symmetric" means and how scalar multiplication interacts with the transpose operation. A matrix is skew-symmetric when its transpose equals its negative. That’s the defining property: $A^T = -A$.

Now, if you take that matrix and multiply every entry by a scalar $k$, you get a new matrix $kA$. The question is: does this new matrix still satisfy the skew-symmetric condition? Let’s check step by step.

1. Start with the definition.

   For $A$ to be skew-symmetric, we must have $A^T = -A$. This is given.

2. Take the transpose of $kA$.

   The transpose of a scalar multiple is the scalar multiple of the transpose. That’s a basic property: $(kA)^T = k A^T$.

   So $(kA)^T = k A^T$.

3. Substitute the skew-symmetric condition.

   Since $A^T = -A$, we get $(kA)^T = k(-A) = -(kA)$.

4. Interpret the result.

   The matrix $kA$ satisfies $(kA)^T = -(kA)$, which is exactly the definition of a skew-symmetric matrix. So $kA$ is skew-symmetric. …

Method: Check the symmetry type under scalar multiplication

Use this to decide whether scaling a matrix keeps it symmetric or skew-symmetric.

Steps

Step 1: State the given symmetry.

Skew-symmetric means $A' = -A$.

Step 2: Transpose the scalar multiple.

A scalar passes straight through the transpose:

$(kA)' = kA' .$ …

Common Mistakes

Mistake 1: Thinking a negative scalar flips it to symmetric.

Why it's wrong: $(kA)' = -(kA)$ holds whatever the sign of $k$, so it stays skew-symmetric. Correct approach: apply $(kA)' = kA'$ and substitute $A'=-A$; the sign of $k$ is irrelevant.

Mistake 2: Answering "symmetric" out of habit. …