Let \( A \) be a matrix given by \( A=[a_{ij}]_{3\times 3} \), \( a_{ij}=i+j \), then
Match List-I with List-II
| List-I | List-II | ||
| (A) | Number of the elements in \( A \) | (I) | 3 |
| (B) | Sum of the diagonal elements | (II) | 9 |
| (C) | \( a_{31}+a_{32}-a_{33} \) | (III) | \( -1 \) |
| (D) | Cofactor of \( a_{13} \) | (IV) | 12 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (II), (B) - (IV), (C) - (I), (D) - (III)
Since \( a_{ij}=i+j \), the matrix is
\( \displaystyle A=\begin{bmatrix}2 & 3 & 4\\3 & 4 & 5\\4 & 5 & 6\end{bmatrix} \)
(A) Number of the elements in A
This matches (II) 9.
A matrix of order 3 × 3 has 3 × 3 = 9 elements.
(B) Sum of the diagonal elements
This matches (IV) 12.
\( \displaystyle a_{11}+a_{22}+a_{33}=2+4+6=12 \)
(C) \( a_{31}+a_{32}-a_{33} \)
This matches (I) 3.
\( \displaystyle a_{31}+a_{32}-a_{33}=4+5-6=3 \)
(D) Cofactor of \( a_{13} \)
This matches (III) \( -1 \).
The cofactor is \( A_{13}=(-1)^{1+3}M_{13} \), where \( M_{13} \) is the determinant left after deleting row 1 and column 3:
\( \displaystyle A_{13}=(+1)\begin{vmatrix}3 & 4\\4 & 5\end{vmatrix} \)
\( \displaystyle =15-16=-1 \)
So the matching is (A)-(II), (B)-(IV), (C)-(I), (D)-(III).