For the determinant \(\begin{vmatrix}5 & -3\\ 2 & 7\end{vmatrix}\), let \(M_{ij}\) and \(A_{ij}\) denote the minor and the cofactor of the element in row i and column j. Consider the following statements:
(A) \(M_{12} = 2\)
(B) \(A_{12} = 2\)
(C) \(A_{21} = 3\)
(D) \(A_{22} = -5\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
In a \(2 \times 2\) determinant, the minor of an element is the single element left after deleting its row and column. The cofactor is \(A_{ij} = (-1)^{i+j}M_{ij}\).
\(M_{11} = 7,\ M_{12} = 2,\ M_{21} = -3,\ M_{22} = 5\)
\(A_{11} = 7,\ A_{12} = -2,\ A_{21} = 3,\ A_{22} = 5\)
So \(M_{12} = 2\), \(A_{12} = -2\), \(A_{21} = 3\) and \(A_{22} = 5\).
Checking the statements
(A) \(M_{12} = 2\) — Correct. Deleting row 1 and column 2 leaves 2.
(B) \(A_{12} = 2\) — Incorrect. \(A_{12} = (-1)^{3} \times 2 = -2\).
(C) \(A_{21} = 3\) — Correct. \(A_{21} = (-1)^{3} \times (-3) = 3\).
(D) \(A_{22} = -5\) — Incorrect. \(A_{22} = (-1)^{4} \times 5 = 5\).