Given that \( [a_{ij}] \) is a square matrix of order n.
Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| A. | \( A=[a_{ij}];\ a_{ij}=\begin{cases}1, & \forall\ i=j\\0, & \forall\ i\ne j\end{cases} \) | I. | Identity Matrix |
| B. | \( B=[a_{ij}];\ a_{ij}=\begin{cases}2i-3j, & \forall\ i=j\\0, & \forall\ i\ne j\end{cases} \) | II. | Diagonal Matrix |
| C. | \( C=[a_{ij}];\ a_{ij}=\begin{cases}k, & \forall\ i=j\\0, & \forall\ i\ne j\end{cases} \) | III. | Scalar Matrix |
| D. | \( D=[a_{ij}];\ a_{ij}=\begin{cases}2i-3j, & \forall\ i\le j\\0, & \forall\ i\gt j\end{cases} \) | 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 - I, B - II, C - III, D - IV
(A) \( A=[a_{ij}];\ a_{ij}=\begin{cases}1, & \forall\ i=j\\0, & \forall\ i\ne j\end{cases} \)
This matches (I) Identity Matrix.
Every diagonal entry is 1 and every other entry is 0.
(B) \( B=[a_{ij}];\ a_{ij}=\begin{cases}2i-3j, & \forall\ i=j\\0, & \forall\ i\ne j\end{cases} \)
This matches (II) Diagonal Matrix.
All off-diagonal entries are 0, and the diagonal entries \( a_{ii}=2i-3i=-i \) are −1, −2, …, −n, which are not all equal, so B is diagonal but not scalar.
(C) \( C=[a_{ij}];\ a_{ij}=\begin{cases}k, & \forall\ i=j\\0, & \forall\ i\ne j\end{cases} \)
This matches (III) Scalar Matrix.
A diagonal matrix with all diagonal entries equal (to k) is a scalar matrix.
(D) \( D=[a_{ij}];\ a_{ij}=\begin{cases}2i-3j, & \forall\ i\le j\\0, & \forall\ i\gt j\end{cases} \)
This matches (IV) Upper Triangular Matrix.
Every entry below the diagonal (i > j) is 0, while entries above it such as \( a_{12}=2-6=-4 \) are not zero.
So the matching is A - I, B - II, C - III, D - IV.
Note: Some List-I items fit more than one List-II entry. The identity matrix in (A) is also a diagonal, a scalar and an upper triangular matrix; the diagonal matrix in (B) is also an upper triangular matrix; and the scalar matrix in (C) is also a diagonal and an upper triangular matrix. The question can still be solved because (D) has only one possible match, (IV), and Option (1) is the only option containing D-IV.