Match List-I with List-II
| List-I | List-II | ||
| (A) | \( \begin{bmatrix}5 & 3\\-7 & 1\end{bmatrix}+\begin{bmatrix}7 & 3\\7 & 2\end{bmatrix} \) | (I) | Identity matrix |
| (B) | \( \begin{bmatrix}5 & 3\\-7 & 1\end{bmatrix}+\begin{bmatrix}-4 & -3\\-6 & 0\end{bmatrix} \) | (II) | Upper Triangular Matrix |
| (C) | \( \begin{bmatrix}5 & -3\\-2 & -1\end{bmatrix}+\begin{bmatrix}-5 & 3\\2 & 1\end{bmatrix} \) | (III) | Zero Matrix |
| (D) | \( \begin{bmatrix}5 & 13\\-2 & 6\end{bmatrix}+\begin{bmatrix}-4 & -13\\2 & -5\end{bmatrix} \) | (IV) | Lower Triangular Matrix |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (II), (B) - (IV), (C) - (III), (D) - (I)
Each sum is found by adding the corresponding entries.
(A) \( \begin{bmatrix}5 & 3\\-7 & 1\end{bmatrix}+\begin{bmatrix}7 & 3\\7 & 2\end{bmatrix} \)
This matches (II) Upper Triangular Matrix. The entry below the diagonal is 0:
\( \displaystyle \begin{bmatrix}5+7 & 3+3\\-7+7 & 1+2\end{bmatrix}=\begin{bmatrix}12 & 6\\0 & 3\end{bmatrix} \)
(B) \( \begin{bmatrix}5 & 3\\-7 & 1\end{bmatrix}+\begin{bmatrix}-4 & -3\\-6 & 0\end{bmatrix} \)
This matches (IV) Lower Triangular Matrix. The entry above the diagonal is 0:
\( \displaystyle \begin{bmatrix}5-4 & 3-3\\-7-6 & 1+0\end{bmatrix}=\begin{bmatrix}1 & 0\\-13 & 1\end{bmatrix} \)
(C) \( \begin{bmatrix}5 & -3\\-2 & -1\end{bmatrix}+\begin{bmatrix}-5 & 3\\2 & 1\end{bmatrix} \)
This matches (III) Zero Matrix:
\( \displaystyle \begin{bmatrix}5-5 & -3+3\\-2+2 & -1+1\end{bmatrix}=\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix} \)
(D) \( \begin{bmatrix}5 & 13\\-2 & 6\end{bmatrix}+\begin{bmatrix}-4 & -13\\2 & -5\end{bmatrix} \)
This matches (I) Identity matrix:
\( \displaystyle \begin{bmatrix}5-4 & 13-13\\-2+2 & 6-5\end{bmatrix}=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} \)
So the matching is (A)-(II), (B)-(IV), (C)-(III), (D)-(I).
Note: Some List-I items fit more than one List-II entry. The zero matrix in (C) and the identity matrix in (D) have zeros both above and below the diagonal, so each of them is also an upper triangular and a lower triangular matrix. The question can still be solved because (A) and (B) have only one possible match each, (A)-(II) and (B)-(IV), and Option (1) is the only option containing them.