Q.If the matrix $\begin{bmatrix}0 & a & 3\\ 2 & b & -1\\ c & 1 & 0\end{bmatrix}$ is a skew symmetric matrix, find the values of $a$, $b$ and $c$.

Concept: Symmetric and Skew-Symmetric Matrices

A square matrix $A$ is skew-symmetric if $A^T = -A$. This forces diagonal entries to be zero and off-diagonal pairs to be negatives of each other.

Let the given matrix be

For $A$ to be skew-symmetric:

1. All diagonal entries must be $0$ — here $0$ and $0$ are fine, so $b = 0$.
2. For $i \neq j$, $a_{ij} = -a_{ji}$. …

The key idea is simple: a matrix is skew-symmetric when its transpose equals its negative. That single condition tells you everything about the entries — no guesswork needed.

For any skew-symmetric matrix $A = [a_{ij}]$, the definition $A^T = -A$ means:

• Every diagonal entry must be zero, because $a_{ii} = -a_{ii}$ implies $2a_{ii} = 0$.
• For off-diagonal entries, $a_{ji} = -a_{ij}$ — the entry at $(j,i)$ is the negative of the entry at $(i,j)$.

Let’s apply this to the given matrix:

1. Diagonal entries must be zero.

   The diagonal entries are $0$, $b$, and $0$. The first and third are already zero, so the condition forces $b = 0$.

2. Compare symmetric pairs across the diagonal.

   Take the $(1,2)$ and $(2,1)$ positions:

   $a_{12} = a$ and $a_{21} = 2$. Skew-symmetry requires $a_{21} = -a_{12}$, so

3. Now check the $(1,3)$ and $(3,1)$ pair: $a_{13} = 3$ and $a_{31} = c$. The condition gives

4. Finally, verify the $(2,3)$ and $(3,2)$ pair as a consistency check: …

Method: Finding unknowns from a symmetry or skew-symmetry condition

When a matrix is stated to be symmetric ($A'=A$) or skew-symmetric ($A'=-A$), that condition pins down its entries. Translate the definition into entry relations and solve for the unknowns.

Steps

Step 1: Write the defining condition entrywise.

Symmetric: $a_{ij}=a_{ji}$ for all $i,j$. Skew-symmetric: $a_{ij}=-a_{ji}$ for all $i,j$.

Step 2: Handle the diagonal.

For a skew-symmetric matrix, $a_{ii}=-a_{ii}$ forces every diagonal entry to be $0$. Set any diagonal unknown to $0$ immediately.

Step 3: Pair up off-diagonal positions. …

Common Mistakes

Mistake 1: Forgetting that the diagonal entries of a skew-symmetric matrix must be zero.

Why it's wrong: skew-symmetry requires $a_{ii}=-a_{ii}$, hence $a_{ii}=0$; leaving a nonzero diagonal unknown gives a wrong value. Correct approach: set the diagonal unknown (here $b$) to $0$.

Mistake 2: Using $a_{ij}=a_{ji}$ (symmetric relation) instead of $a_{ij}=-a_{ji}$.

Why it's wrong: for skew-symmetry the mirror entries are negatives, so $a=-2$ and $c=-3$, not $+2$ and $+3$. Correct approach: apply the minus sign from $A'=-A$. …