Match List-I with List-II
| List-I | List-II | ||
| (A) | Determinant of a skew symmetric matrix of order 3 is | (I) | 1 |
| (B) | If A & B are square matrices of same order 3 such that 3AB = I, I is an identity matrix of order 3 & |B| = 1/54 then |A| = | (II) | 0 |
| (C) | In the matrix \( A=\begin{bmatrix}1 & 2\\ -1 & 3\end{bmatrix} \) the ratio of cofactor to minor of element 3 is | (III) | 9 |
| (D) | If \( P=\begin{bmatrix}1 & x & 4\\ 1 & 4 & 4\\ 2 & 3 & 3\end{bmatrix} \) is adjoint of 3 x 3 matrix A & |A| = 5 then \( x \) is | (IV) | 2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (II), (B) - (IV), (C) - (I), (D) - (III)
(A) Determinant of a skew symmetric matrix of order 3
This matches (II).
For a skew symmetric matrix, \( A^{T}=-A \), so \( |A|=|A^{T}|=(-1)^{3}|A|=-|A| \), which gives \( |A|=0 \).
(B) \( |A| \) when \( 3AB=I \) and \( |B|=\frac{1}{54} \)
This matches (IV).
Taking determinants in \( 3AB=I \) gives \( 3^{3}|A||B|=1 \):
\( \displaystyle |A|=\frac{1}{27\times\frac{1}{54}} \)
\( =2 \)
(C) Ratio of cofactor to minor of the element 3
This matches (I).
The element 3 is \( a_{22} \). Its minor is 1 and its cofactor is \( (-1)^{2+2}\cdot 1=1 \), so the ratio is 1.
(D) \( x \) when \( P=\mathrm{adj}\,A \) and \( |A|=5 \)
This matches (III).
Since \( P=\mathrm{adj}\,A \), we have \( |P|=|A|^{2}=25 \). Expanding \( |P| \) along the first row:
\( |P|=1(12-12)-x(3-8)+4(3-8) \)
\( =5x-20 \)
Setting \( 5x-20=25 \) gives \( x=9 \).
So the matching is (A)-(II), (B)-(IV), (C)-(I), (D)-(III).