Q.The matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is a
(A) identity matrix
(B) symmetric matrix
(C) skew symmetric matrix
(D) none of these

The key idea is that a symmetric matrix equals its own transpose ($A^T = A$), while a skew-symmetric matrix satisfies $A^T = -A$.

Step 1: Write the given matrix:

Step 2: Compute its transpose (swap rows and columns). Since it is diagonal, the transpose is identical:

$A^T = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix} = A$ …

Let’s understand why. A matrix is symmetric if it equals its own transpose — that is, $A = A^T$. For a skew-symmetric matrix, $A^T = -A$, which forces all diagonal entries to be zero. The identity matrix is a special diagonal matrix with all diagonal entries equal to 1.

The given matrix is diagonal: all non-diagonal entries are zero. When you take its transpose, the diagonal stays the same and zeros remain zeros. So $A^T = A$ automatically. That’s the core reason — any diagonal matrix is symmetric.

Now check each option:

1. Is it an identity matrix?

   The identity matrix $I_3$ has 1’s on the diagonal. Here the diagonal entries are $1, 2, 4$ — not all 1’s. So no.

2. Is it symmetric?

   As argued, $A^T = A$ because swapping rows and columns leaves a diagonal matrix unchanged. Yes, it is symmetric.

3. Is it skew-symmetric?

   For skew-symmetry, we need $A^T = -A$. That would require $1 = -1$, $2 = -2$, $4 = -4$ on the diagonal — impossible unless all diagonal entries are zero. So no. …

Method: Classify a matrix by comparing it with its transpose

Use this to decide whether a given square matrix is identity, scalar, diagonal, symmetric, or skew-symmetric.

Steps

Step 1: Form the transpose $A'$.

Swap rows and columns. For a diagonal matrix the transpose is the matrix itself, which is worth spotting early.

Step 2: Test the two symmetry conditions.

Compare:

A skew-symmetric matrix must have every diagonal entry $0$, so a non-zero diagonal immediately rules that out. …

Common Mistakes

Mistake 1: Calling any diagonal matrix the "identity matrix".

Why it's wrong: the identity requires every diagonal entry to be exactly $1$; a diagonal such as $\operatorname{diag}(1,2,4)$ is not the identity. Correct approach: check the actual diagonal values before choosing identity.

Mistake 2: Assuming a diagonal matrix cannot be "symmetric" because it looks like its own separate category. …