Let \(A = \begin{bmatrix}1 & 2 & 1\\ -1 & 3 & 2\\ 2 & 4 & 1\end{bmatrix}\) and \(M_{ij}, A_{ij}\) respectively denote the minor, co-factor of an element \(a_{ij}\) of matrix A, then which of the following are true?
(A) \(M_{22} = -1\)
(B) \(A_{23} = 0\)
(C) \(A_{32} = 3\)
(D) \(M_{23} = 1\)
(E) \(M_{32} = -3\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
The minor \(M_{ij}\) is the determinant left after deleting row i and column j, and the cofactor is \(A_{ij} = (-1)^{i+j}M_{ij}\).
(A) \(M_{22} = -1\) — Correct. Deleting row 2 and column 2:
\(\displaystyle M_{22} = \begin{vmatrix}1 & 1\\ 2 & 1\end{vmatrix} = 1 - 2 = -1\)
(B) \(A_{23} = 0\) — Correct. Deleting row 2 and column 3:
\(\displaystyle M_{23} = \begin{vmatrix}1 & 2\\ 2 & 4\end{vmatrix} = 4 - 4 = 0\)
\(A_{23} = (-1)^{5}(0) = 0\)
(C) \(A_{32} = 3\) — Incorrect. Deleting row 3 and column 2:
\(\displaystyle M_{32} = \begin{vmatrix}1 & 1\\ -1 & 2\end{vmatrix} = 2 + 1 = 3\)
\(A_{32} = (-1)^{5}(3) = -3\)
(D) \(M_{23} = 1\) — Incorrect. \(M_{23} = 4 - 4 = 0\).
(E) \(M_{32} = -3\) — Incorrect. \(M_{32} = 2 + 1 = 3\); −3 is the cofactor \(A_{32}\).