Let \(A = \begin{bmatrix}2 & -1 & 3\\ 1 & 4 & 0\\ -2 & 1 & 5\end{bmatrix}\), and let \(C_{ij}\) denote the cofactor of the element \(a_{ij}\) of A.
Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | \(C_{11}\) | (I) | \(-5\) |
| (B) | \(C_{12}\) | (II) | 8 |
| (C) | \(C_{21}\) | (III) | 9 |
| (D) | \(C_{33}\) | (IV) | 20 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(IV), (B)-(I), (C)-(II), (D)-(III)
The cofactor is \(C_{ij} = (-1)^{i+j}M_{ij}\), where the minor \(M_{ij}\) is the determinant left after deleting row i and column j.
(A) \(C_{11}\)
This matches (IV) 20.
\(C_{11} = +\big[(4)(5) - (0)(1)\big] = 20\)
(B) \(C_{12}\)
This matches (I) \(-5\).
\(C_{12} = -\big[(1)(5) - (0)(-2)\big] = -5\)
(C) \(C_{21}\)
This matches (II) 8.
\(C_{21} = -\big[(-1)(5) - (3)(1)\big] = -(-8) = 8\)
(D) \(C_{33}\)
This matches (III) 9.
\(C_{33} = +\big[(2)(4) - (-1)(1)\big] = 9\)