If a matrix A is both symmetric and skew-symmetric, then: - Vidyadip

MCQ

If a matrix $A$ is both symmetric and skew-symmetric, then:

A
$A$ is a diagonal matrix.
✓
$A$ is a zero matrix.
C
$A$ is a scalar matrix.
D
$A$ is a square matrix.

✓

Answer

Correct option: B.

$A$ is a zero matrix.

$A$ is symmetric $\Rightarrow a _{ ij }= a _{ ji } \rightarrow$ (1)$A$ is skew$-$symmetric
$\Rightarrow a_{i j}=-a_{i j} \rightarrow(2)$ and
$a_{i j}=-a_{i j}$
$\Rightarrow a_{i j}=0$ means the diagonal entries are zero.
From $(1)$ and $(2)$ we can write $a_{i j}=a_{i j}=0$ which means all the off diagonal entries are zero.
So, $A$ is a null matrix.

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