If \( A=\begin{bmatrix}3 & 2 & a\\ b & 1 & c\\ -2 & -1 & 0\end{bmatrix} \) is symmetric matrix, then which of the following is/are correct?
(A) \( a=2,\ b=-2,\ c=1 \)
(B) \( a_{21}+a_{32}=1 \) (Where \( a_{ij} \) is \( (i,j)^{th} \) element of matrix \( A \))
(C) Diagonal elements are -2, 1, -2
(D) \( 2a+3b-c=3 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (D) only
For a symmetric matrix, \( a_{ij}=a_{ji} \). Comparing the entries placed symmetrically about the diagonal:
\( a_{12}=a_{21}\Rightarrow b=2 \)
\( a_{13}=a_{31}\Rightarrow a=-2 \)
\( a_{23}=a_{32}\Rightarrow c=-1 \)
So \( a=-2 \), \( b=2 \), \( c=-1 \), and the diagonal elements are 3, 1 and 0.
Checking the statements
(A) \( a=2,\ b=-2,\ c=1 \) — Incorrect. Every sign is reversed.
(B) \( a_{21}+a_{32}=1 \) — Correct. \( a_{21}+a_{32}=2+(-1)=1 \).
(C) Diagonal elements are -2, 1, -2 — Incorrect. The diagonal elements are 3, 1 and 0.
(D) \( 2a+3b-c=3 \) — Correct. \( 2(-2)+3(2)-(-1)=-4+6+1=3 \).