Let \( A=[a_{ij}]_{3\times 3} \) be a matrix of order 3,then
Match List-I with List-II
| List-I | List-II | ||
| (A) | \( a_{ij}=\begin{cases}\displaystyle\frac{5j}{2i}, & \forall\ i=j\\ 0, & \forall\ i\ne j\end{cases} \) | (I). | Lower Triangular Matrix |
| (B). | \( a_{ij}=\begin{cases}2i+3j, & \forall\ i\ge j\\ 0, & \forall\ i\lt j\end{cases} \) | (II). | Identity Matrix |
| (C) | \( a_{ij}=\begin{cases}i^{2}+j^{2}, & \forall\ i\le j\\ 0, & \forall\ i\gt j\end{cases} \) | (III). | Scalar Matrix |
| (D) | \( a_{ij}=\begin{cases}\displaystyle\frac{i}{j}, & \forall\ i=j\\ 0, & \forall\ i\ne j\end{cases} \) | (IV). | Upper Triangular Matrix |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (III), (B) - (I), (C) - (IV), (D) - (II)
(A) \( a_{ij}=\frac{5j}{2i} \) for \( i=j \), 0 otherwise
This matches (III) Scalar Matrix.
The diagonal entries are \( \frac{5i}{2i}=\frac{5}{2} \), all equal, and every other entry is 0, so it is a scalar matrix.
(B) \( a_{ij}=2i+3j \) for \( i\ge j \), 0 for \( i\lt j \)
This matches (I) Lower Triangular Matrix.
Every entry above the diagonal is 0, while entries below it, such as \( a_{21}=7 \), are not, so it is lower triangular.
(C) \( a_{ij}=i^{2}+j^{2} \) for \( i\le j \), 0 for \( i\gt j \)
This matches (IV) Upper Triangular Matrix.
Every entry below the diagonal is 0, while entries above it, such as \( a_{12}=5 \), are not, so it is upper triangular.
(D) \( a_{ij}=\frac{i}{j} \) for \( i=j \), 0 otherwise
This matches (II) Identity Matrix.
The diagonal entries are \( \frac{i}{i}=1 \) and every other entry is 0, so it is the identity matrix.
So the matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).
Note: Some List-I items fit more than one List-II entry. The scalar matrix in (A) also fits (I) and (IV), since it is both lower triangular and upper triangular, and the identity matrix in (D) fits (I), (III) and (IV) as well as (II). The question can still be solved because (B) and (C) have only one possible match each, (B)-(I) and (C)-(IV), and Option (1) is the only option containing them.