The sum of two skew-symmetric matrices of the same order is a
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → skew-symmetric matrix
Let A and B be skew-symmetric, so \( \mathrm{A}^{\mathrm{T}}=-\mathrm{A} \) and \( \mathrm{B}^{\mathrm{T}}=-\mathrm{B} \). Then
\( (\mathrm{A}+\mathrm{B})^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\mathrm{B}^{\mathrm{T}} \)
\( =-\mathrm{A}-\mathrm{B} \)
\( =-(\mathrm{A}+\mathrm{B}) \)
So A + B is a skew-symmetric matrix.
Why the other options are incorrect
Option (1) symmetric matrix — Incorrect. A + B is symmetric only in the special case A + B = O; for example \( \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}+\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}=\begin{bmatrix}0 & 2\\-2 & 0\end{bmatrix} \) is not symmetric.
Option (2) zero matrix — Incorrect. The sum is the zero matrix only when B = −A; the example in Option (1) gives a non-zero sum.
Option (4) diagonal matrix — Incorrect. The sum \( \begin{bmatrix}0 & 2\\-2 & 0\end{bmatrix} \) has non-zero off-diagonal entries, so the sum need not be diagonal.
NCERT Exemplar:
Matrices
