If \( a_{ij} \) and \( A_{ij} \) represents the elements and cofactors of the matrix \( A=\begin{bmatrix}2 & 1 & 3\\3 & 2 & 1\\1 & 2 & 3\end{bmatrix} \) then which of the following statements are True?
(A) \( |A| \) = 28
(B) \( A_{23} \) = -3
(C) \( a_{11}\,A_{12} \) = -16
(D) \( \displaystyle\frac{A_{21}}{A_{13}}=\frac{3}{4} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
The cofactor is \( A_{ij}=(-1)^{i+j}M_{ij} \), where \( M_{ij} \) is the minor obtained by deleting row i and column j. The cofactors needed are:
\( \displaystyle A_{11}=\begin{vmatrix}2 & 1\\2 & 3\end{vmatrix}=4 \)
\( \displaystyle A_{12}=-\begin{vmatrix}3 & 1\\1 & 3\end{vmatrix}=-8 \)
\( \displaystyle A_{13}=\begin{vmatrix}3 & 2\\1 & 2\end{vmatrix}=4 \)
\( \displaystyle A_{21}=-\begin{vmatrix}1 & 3\\2 & 3\end{vmatrix}=-(3-6)=3 \)
\( \displaystyle A_{23}=-\begin{vmatrix}2 & 1\\1 & 2\end{vmatrix}=-(4-1)=-3 \)
Expanding along the first row:
\( \displaystyle |A|=a_{11}A_{11}+a_{12}A_{12}+a_{13}A_{13} \)
\( \displaystyle =2(4)+1(-8)+3(4) \)
\( \displaystyle =12 \)
So |A| = 12, \( A_{12}=-8 \), \( A_{13}=4 \), \( A_{21}=3 \) and \( A_{23}=-3 \).
Checking the statements
(A) \( |A| \) = 28 — Incorrect. |A| is 12.
(B) \( A_{23} \) = -3 — Correct. \( A_{23}=-(4-1)=-3 \).
(C) \( a_{11}\,A_{12} \) = -16 — Correct. \( 2\times(-8)=-16 \).
(D) \( \frac{A_{21}}{A_{13}}=\frac{3}{4} \) — Correct. \( A_{21}=3 \) and \( A_{13}=4 \).