Let \(A=\begin{bmatrix}2&-3&4\\0&1&5\\-4&2&3\end{bmatrix}\), and \(a_{ij}\) be any element of matrix A, \(i, j \in \{1,2,3\}\), then which of the following are TRUE?
(A) Minor of \(a_{23} = 16\)
(B) Minor of \(a_{23} = -8\)
(C) Cofactor of \(a_{23} = -16\)
(D) Cofactor of \(a_{23} = 8\)
(E) Cofactor of \(a_{13} = 4\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B), (D) and (E) only
The minor \(M_{ij}\) is the determinant left after deleting row i and column j. The cofactor is \(A_{ij} = (-1)^{i+j}M_{ij}\).
Deleting row 2 and column 3:
\( M_{23} = \begin{vmatrix}2&-3\\-4&2\end{vmatrix} = 4 - 12 = -8 \)
\(A_{23} = (-1)^{5}(-8) = 8\)
Deleting row 1 and column 3:
\( M_{13} = \begin{vmatrix}0&1\\-4&2\end{vmatrix} = 0 + 4 = 4 \)
\(A_{13} = (-1)^{4}(4) = 4\)
So \(M_{23} = -8\), \(A_{23} = 8\) and \(A_{13} = 4\).
Checking the statements
(A) Minor of \(a_{23} = 16\) — Incorrect. \(M_{23} = -8\).
(B) Minor of \(a_{23} = -8\) — Correct. Deleting row 2 and column 3 gives \(M_{23} = 4 - 12 = -8\).
(C) Cofactor of \(a_{23} = -16\) — Incorrect. \(A_{23} = 8\).
(D) Cofactor of \(a_{23} = 8\) — Correct. \(i + j = 5\) is odd, so the sign of the minor changes.
(E) Cofactor of \(a_{13} = 4\) — Correct. \(i + j = 4\) is even, so \(A_{13} = M_{13} = 4\).