If matrix \( A=\begin{bmatrix}5 & 2 & 3\\ 4 & 1 & 2\\ 7 & 6 & 0\end{bmatrix} \), then
Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| A. | \( |A| \) | I. | 16 |
| B. | Minor of \( a_{23} \) | II. | 27 |
| C. | Cofactor of \( a_{32} \) | III. | 19 |
| D. | If \( a_{23} \) is interchanged with \( a_{32} \), then \( |A| \) | IV. | 2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A-III, B-I, C-IV, D-II
(A) Expanding \( |A| \) along the first row,
\( \displaystyle |A|=5(1\cdot0-2\cdot6)-2(4\cdot0-2\cdot7)+3(4\cdot6-1\cdot7) \)
\( \displaystyle =-60+28+51 \)
\( \displaystyle =19 \)
This matches (III).
(B) Deleting row 2 and column 3 gives the minor of \( a_{23} \).
\( \displaystyle M_{23}=\begin{vmatrix}5&2\\7&6\end{vmatrix} \)
\( \displaystyle =30-14 \)
\( \displaystyle =16 \)
This matches (I).
(C) Deleting row 3 and column 2 gives the minor of \( a_{32} \).
\( \displaystyle M_{32}=\begin{vmatrix}5&3\\4&2\end{vmatrix} \)
\( \displaystyle =10-12 \)
\( \displaystyle =-2 \)
The cofactor is
\( \displaystyle C_{32}=(-1)^{3+2}M_{32} \)
\( \displaystyle =(-1)(-2) \)
\( \displaystyle =2 \)
This matches (IV).
(D) Interchanging \( a_{23}=2 \) and \( a_{32}=6 \) gives
\( \displaystyle A'=\begin{bmatrix}5&2&3\\4&1&6\\7&2&0\end{bmatrix} \)
\( \displaystyle |A'|=5(0-12)-2(0-42)+3(8-7) \)
\( \displaystyle =-60+84+3 \)
\( \displaystyle =27 \)
This matches (II).
So the matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).