Let \(A = \begin{bmatrix}1 & 2 & -1\\ 3 & 0 & 2\\ 2 & 1 & 4\end{bmatrix} = [a_{ij}]\) and let \(C_{ij}\) be the cofactor of \(a_{ij}\). Consider the following statements:
(A) \(a_{21}C_{21} + a_{22}C_{22} + a_{23}C_{23} = -21\)
(B) \(a_{11}C_{21} + a_{12}C_{22} + a_{13}C_{23} = 0\)
(C) \(a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} = 21\)
(D) \(a_{11}C_{12} + a_{21}C_{22} + a_{31}C_{32} = -21\)
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 elements of a row multiplied by their own cofactors and added give \(|A|\). The elements of a row (or column) multiplied by the cofactors of another row (or column) and added give 0.
Expanding along the first row:
\(|A| = 1(0 - 2) - 2(12 - 4) + (-1)(3 - 0)\)
\(|A| = -2 - 16 - 3\)
\(|A| = -21\)
The cofactors of the second row are:
\(C_{21} = -(8 + 1) = -9,\ \ C_{22} = 4 + 2 = 6,\ \ C_{23} = -(1 - 4) = 3\)
So \(|A| = -21\), and the second-row cofactors are \(-9, 6, 3\).
Checking the statements
(A) \(a_{21}C_{21} + a_{22}C_{22} + a_{23}C_{23} = -21\) — Correct. \(3(-9) + 0(6) + 2(3) = -21 = |A|\).
(B) \(a_{11}C_{21} + a_{12}C_{22} + a_{13}C_{23} = 0\) — Correct. \(1(-9) + 2(6) + (-1)(3) = 0\).
(C) \(a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} = 21\) — Incorrect. This sum is \(|A| = -21\), not 21.
(D) \(a_{11}C_{12} + a_{21}C_{22} + a_{31}C_{32} = -21\) — Incorrect. Column-1 elements with column-2 cofactors give \(1(-8) + 3(6) + 2(-5) = 0\).