Match List-I with List-II
| List-I | List-II | ||
| (A) | If \( A=\left[a_{ij}\right]_{n\times n} \), where \( a_{ij}=\begin{cases}0, & i\ne j\\k, & i=j,k\ne 0\end{cases} \) then \( A \) is ___ | (I) | Symmetric matrix |
| (B) | If \( A=\left[a_{ij}\right]_{2\times 2} \), where \( a_{ij}=\begin{cases}1, & i\ne j\\0, & i=j\end{cases} \), then \( A^2 \) is ___ | (II) | Skew-Symmetric matrix |
| (C) | If \( A \) and \( B \) are matrices of the same order, then \( (AB^T-BA^T) \) is ___ | (III) | Identity matrix |
| (D) | If \( A \) and \( B \) are symmetric matrices of the same order, then \( (AB+BA) \) is ___ | (IV) | Scalar matrix |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (IV), (B) - (III), (C) - (II), (D) - (I)
(A) \( a_{ij}=0 \) for \( i\ne j \) and \( a_{ii}=k \), \( k\ne 0 \)
This matches (IV) Scalar matrix. All the off-diagonal entries are 0 and all the diagonal entries are the same number k, which is the definition of a scalar matrix.
(B) \( A=\left[a_{ij}\right]_{2\times 2} \) with \( a_{ij}=1 \) for \( i\ne j \) and 0 for \( i=j \)
This matches (III) Identity matrix.
\( \displaystyle A=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix} \)
\( \displaystyle A^2=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}=\mathrm{I} \)
(C) \( AB^T-BA^T \)
This matches (II) Skew-Symmetric matrix. Using \( (PQ)^T=Q^TP^T \) and \( (P^T)^T=P \):
\( \displaystyle (AB^T-BA^T)^T=(B^T)^TA^T-(A^T)^TB^T \)
\( \displaystyle =BA^T-AB^T \)
\( \displaystyle =-(AB^T-BA^T) \)
(D) \( AB+BA \), with \( A^T=A \) and \( B^T=B \)
This matches (I) Symmetric matrix.
\( \displaystyle (AB+BA)^T=B^TA^T+A^TB^T \)
\( \displaystyle =BA+AB \)
\( \displaystyle =AB+BA \)
So the matching is (A)-(IV), (B)-(III), (C)-(II), (D)-(I).
Note: Some List-I items fit more than one List-II entry. The scalar matrix in (A) is also symmetric, so (A) fits (I) and (IV); the identity matrix in (B) is also symmetric and scalar, so (B) fits (I), (III) and (IV). The question can still be solved because (C) fits only (II) and (D) fits only (I), which leaves Options (3) and (4). Option (3) matches (A) with (III), but the matrix in (A) is the identity matrix only when k = 1, so Option (4) is the only option in which every item has a valid match.