Q.The matrix $\begin{bmatrix} 0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0 \end{bmatrix}$ is a
(A) diagonal matrix
(B) symmetric matrix
(C) skew symmetric matrix
(D) scalar matrix

The key idea is that a matrix $A$ is skew symmetric if $A^T = -A$, meaning the transpose equals the negative of the original.

Step 1: Write the given matrix $A$ and compute its transpose $A^T$ (swap rows and columns):

Step 2: Compare $A^T$ with $-A$:

We see $A^T = -A$ exactly. …

Why this approach works

The question gives you a $3 \times 3$ matrix and asks you to classify it among four types: diagonal, symmetric, skew-symmetric, or scalar. Instead of memorising definitions in isolation, think about what each type does to the entries.

A symmetric matrix is unchanged when you flip it across the main diagonal — that means $a_{ij} = a_{ji}$ for every pair. A skew-symmetric matrix, on the other hand, flips sign when you transpose it: $a_{ij} = -a_{ji}$. And a key consequence? The diagonal entries of a skew-symmetric matrix must be zero, because $a_{ii} = -a_{ii}$ forces $a_{ii}=0$.

Look at the given matrix:

The diagonal is all zeros — that already rules out a diagonal or scalar matrix (which would need non-zero entries on the diagonal, or at least a constant $k$ there). So the real contest is between symmetric and skew-symmetric. Let’s check systematically.

Step-by-step reasoning

1. Check the diagonal entries.

   For a symmetric matrix, diagonal entries can be anything. For a skew-symmetric matrix, they must be zero. Here $a_{11}=0$, $a_{22}=0$, $a_{33}=0$ — so the diagonal condition for skew-symmetry is satisfied. But this alone isn’t enough; we need to check the off-diagonal pairs.

2. Compare each pair $(i,j)$ and $(j,i)$.

   Take $a_{12} = -5$ and $a_{21} = 5$.

   Is $a_{12} = -a_{21}$? Yes: $-5 = -(5)$.

   Next, $a_{13} = 8$ and $a_{31} = -8$.

   $8 = -(-8)$ — again true.

   Finally, $a_{23} = 12$ and $a_{32} = -12$.

   $12 = -(-12)$ — holds.

   Every off-diagonal pair satisfies $a_{ij} = -a_{ji}$. That is the defining property of a skew-symmetric matrix.

3. Rule out the other options.

   • Diagonal matrix: would have all non-diagonal entries zero. Here $-5, 8, 5, 12, -8, -12$ are all non-zero — so no.
   • Symmetric matrix: would require $a_{ij} = a_{ji}$. But $-5 \neq 5$, $8 \neq -8$, $12 \neq -12$ — so no.
   • Scalar matrix: a diagonal matrix where every diagonal entry is the same constant $k$. Here the diagonal is all zeros, so $k=0$ — but then all off-diagonals must also be zero (since a scalar matrix is a special diagonal matrix). They aren’t — so no. …

Method: Identify a skew-symmetric matrix

Use this to test whether a square matrix is skew-symmetric (and to distinguish it from diagonal, symmetric, or scalar matrices).

Steps

Step 1: Scan the main diagonal.

Skew-symmetry requires $a_{ii} = -a_{ii}$, hence every diagonal entry must be $0$. If any diagonal entry is non-zero, the matrix cannot be skew-symmetric (and this also rules out diagonal/scalar types, which need non-zero diagonals).

Step 2: Check each mirror pair across the diagonal.

For every pair test …

Common Mistakes

Mistake 1: Concluding skew-symmetric from a zero diagonal alone.

Why it's wrong: a zero diagonal is necessary but not sufficient — e.g. $\begin{bmatrix}0&1\\2&0\end{bmatrix}$ has zero diagonal yet $1 \neq -2$, so it is not skew-symmetric. Correct approach: also verify $a_{ij} = -a_{ji}$ for every off-diagonal pair.

Mistake 2: Calling it a diagonal matrix because the diagonal is all zeros. …