Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| A. | A matrix which is both symmetric and skew symmetric | I. | symmetric matrix |
| B. | All positive integral powers of a symmetric matrix | II. | unit matrix |
| C. | Positive odd integral powers of a skew-symmetric matrix | III. | null matrix |
| D. | A square matrix \( A=[a_{ij}]_{n\times n} \) with \( a_{ij}=\begin{cases}0, & \forall i\ne j\\ 1, & \forall i=j\end{cases} \) | IV. | skew-symmetric matrix |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A-III, B-I, C-IV, D-II
(A) A matrix which is both symmetric and skew symmetric
This matches (III) null matrix. With \( A^{T}=A \) and \( A^{T}=-A \):
\( A=-A \)
\( 2A=O,\quad A=O \)
(B) All positive integral powers of a symmetric matrix
This matches (I) symmetric matrix. With \( A^{T}=A \):
\( (A^{n})^{T}=(A^{T})^{n}=A^{n} \)
(C) Positive odd integral powers of a skew-symmetric matrix
This matches (IV) skew-symmetric matrix. With \( A^{T}=-A \) and n odd:
\( (A^{n})^{T}=(A^{T})^{n}=(-A)^{n}=-A^{n} \)
(D) A square matrix with \( a_{ij}=0 \) for i ≠ j and \( a_{ij}=1 \) for i = j
This matches (II) unit matrix: 1 on the diagonal and 0 elsewhere is the definition of the identity (unit) matrix.
So the matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).