Let \(A = \begin{bmatrix}2&-1&3\\1&2&-1\\4&1&2\end{bmatrix}\). \(M_{ij}\) and \(A_{ij}\) respectively denote the minor and cofactor of an element \(a_{ij}\) of matrix \(A = [a_{ij}]\).
(A) \(M_{23} = 6\)
(B) \(A_{22} = -8\)
(C) \(A_{13} = 7\)
(D) \(M_{32} = -5\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
The minor \(M_{ij}\) is the determinant left after deleting row i and column j, and the cofactor is \(A_{ij} = (-1)^{i+j}M_{ij}\).
\(M_{23}\): delete row 2 and column 3.
\(\displaystyle M_{23} = \begin{vmatrix}2&-1\\4&1\end{vmatrix} = 2 + 4 = 6\)
\(A_{22}\): delete row 2 and column 2; the sign is \((-1)^{4} = +1\).
\(\displaystyle M_{22} = \begin{vmatrix}2&3\\4&2\end{vmatrix} = 4 - 12 = -8\)
\(A_{22} = -8\)
\(A_{13}\): delete row 1 and column 3; the sign is \((-1)^{4} = +1\).
\(\displaystyle M_{13} = \begin{vmatrix}1&2\\4&1\end{vmatrix} = 1 - 8 = -7\)
\(A_{13} = -7\)
\(M_{32}\): delete row 3 and column 2.
\(\displaystyle M_{32} = \begin{vmatrix}2&3\\1&-1\end{vmatrix} = -2 - 3 = -5\)
So \(M_{23} = 6\), \(A_{22} = -8\), \(A_{13} = -7\) and \(M_{32} = -5\).
Checking the statements
(A) \(M_{23} = 6\) — Correct. \(M_{23}\) is 6.
(B) \(A_{22} = -8\) — Correct. \(A_{22}\) is −8.
(C) \(A_{13} = 7\) — Incorrect. \(A_{13}\) is −7.
(D) \(M_{32} = -5\) — Correct. \(M_{32}\) is −5.