Match List - I with List - II.
| List - I | List - II | ||
|---|---|---|---|
| Matrix | Types of matrix | ||
| (A) | \(\begin{bmatrix}1 & 2 & 3\\0 & 5 & 7\\0 & 0 & 4\end{bmatrix}\) | (I) | Skew symmetric matrix |
| (B) | \(\begin{bmatrix}1 & 3 & 5\\3 & 0 & 4\\5 & 4 & 2\end{bmatrix}\) | (II) | Upper triangular matrix |
| (C) | \(\begin{bmatrix}0 & -2 & 1\\2 & 0 & 5\\-1 & -5 & 0\end{bmatrix}\) | (III) | Symmetric matrix |
| (D) | \(\begin{bmatrix}2 & 0 & 0\\0 & 1 & 0\\0 & 0 & 4\end{bmatrix}\) | (IV) | Diagonal matrix |
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
Each matrix is matched with the most specific type it belongs to.
(A) The matrix with rows (1, 2, 3), (0, 5, 7), (0, 0, 4)
This matches (II) Upper triangular matrix: every entry below the diagonal is 0, while some entries above it are not 0. It is not symmetric, since \(a_{12} = 2\) but \(a_{21} = 0\).
(B) The matrix with rows (1, 3, 5), (3, 0, 4), (5, 4, 2)
This matches (III) Symmetric matrix: \(a_{12} = a_{21} = 3\), \(a_{13} = a_{31} = 5\), \(a_{23} = a_{32} = 4\), so \(B^T = B\).
(C) The matrix with rows (0, −2, 1), (2, 0, 5), (−1, −5, 0)
This matches (I) Skew symmetric matrix: the diagonal entries are 0 and \(a_{21} = -a_{12} = 2\), \(a_{31} = -a_{13} = -1\), \(a_{32} = -a_{23} = -5\), so \(C^T = -C\).
(D) diag(2, 1, 4)
This matches (IV) Diagonal matrix: every entry off the diagonal is 0.
Note: Some List-I items fit more than one List-II entry. The diagonal matrix (D) is also a symmetric matrix (III) and an upper triangular matrix (II). The question can still be solved because (A) has only one possible match, (II), (B) has only one possible match, (III), and (C) has only one possible match, (I); Option (4) is the only option containing (A)-(II).