Let \(A = [a_{ij}]\) be a square matrix of order 3. Match List-I (the rule for \(a_{ij}\)) with List-II (the kind of matrix A is).
| List-I | List-II | ||
|---|---|---|---|
| (A) | \(a_{ij} = i + j\) | (I) | Scalar matrix |
| (B) | \(a_{ij} = i^2 - j^2\) | (II) | Skew-symmetric matrix |
| (C) | \(a_{ij} = i\) when \(i = j\), and \(a_{ij} = 0\) when \(i \ne j\) | (III) | Diagonal matrix but not a scalar matrix |
| (D) | \(a_{ij} = 5\) when \(i = j\), and \(a_{ij} = 0\) when \(i \ne j\) | (IV) | Symmetric matrix but not a diagonal matrix |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
A is symmetric when \(a_{ji} = a_{ij}\) and skew-symmetric when \(a_{ji} = -a_{ij}\) for all i and j.
(A) \(a_{ij} = i + j\)
This matches (IV) Symmetric matrix but not a diagonal matrix.
\(a_{ji} = j + i = a_{ij}\), and \(a_{12} = 3 \ne 0\)
(B) \(a_{ij} = i^2 - j^2\)
This matches (II) Skew-symmetric matrix.
\(a_{ji} = j^2 - i^2 = -(i^2 - j^2) = -a_{ij}\)
(C) \(a_{ij} = i\) when \(i = j\), and 0 otherwise
This matches (III) Diagonal matrix but not a scalar matrix.
\( \displaystyle A = \begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix} \), with unequal diagonal elements
(D) \(a_{ij} = 5\) when \(i = j\), and 0 otherwise
This matches (I) Scalar matrix.
\( \displaystyle A = \begin{bmatrix}5&0&0\\0&5&0\\0&0&5\end{bmatrix} = 5I \)