Which of the following are correct?
(A) \( A=\begin{bmatrix}-9 & 4 & -3\\-1 & 0 & 4\\2 & 2 & 0\end{bmatrix} \); \( a_{22}+a_{32}=-9 \) where \( a_{ij} \) denotes the \( (i,j)^{th} \) element of matrix \( A \).
(B) \( \mathrm{A}=\begin{bmatrix}2 & -2\\-2 & 2\end{bmatrix} \); A2 = pA, then p = 4
(C) \( \begin{bmatrix}8 & 6\\6 & 16\end{bmatrix} \) is a symmetric matrix
(D) \( \begin{bmatrix}0 & -5 & 3\\5 & 0 & -7\\-3 & 7 & 0\end{bmatrix} \) is a symmetric matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) and (C) only
In (A), \( a_{22} \) is in row 2, column 2, and \( a_{32} \) in row 3, column 2:
\( \displaystyle a_{22}+a_{32}=0+2=2 \)
In (B),
\( \displaystyle A^2=\begin{bmatrix}2 & -2\\-2 & 2\end{bmatrix}\begin{bmatrix}2 & -2\\-2 & 2\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}4+4 & -4-4\\-4-4 & 4+4\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}8 & -8\\-8 & 8\end{bmatrix} \)
\( \displaystyle =4A \)
A matrix is symmetric when \( a_{ij}=a_{ji} \) for every pair.
So \( a_{22}+a_{32}=2 \), \( A^2=4A \) (p = 4), the 2 × 2 matrix in (C) has \( a_{12}=a_{21}=6 \), and the 3 × 3 matrix in (D) has \( a_{12}=-5 \) but \( a_{21}=5 \).
Checking the statements
(A) \( A=\begin{bmatrix}-9 & 4 & -3\\-1 & 0 & 4\\2 & 2 & 0\end{bmatrix} \); \( a_{22}+a_{32}=-9 \) where \( a_{ij} \) denotes the \( (i,j)^{th} \) element of matrix \( A \). — Incorrect. \( a_{22}+a_{32}=0+2=2 \).
(B) \( \mathrm{A}=\begin{bmatrix}2 & -2\\-2 & 2\end{bmatrix} \); A2 = pA, then p = 4 — Correct. \( A^2=4A \).
(C) \( \begin{bmatrix}8 & 6\\6 & 16\end{bmatrix} \) is a symmetric matrix — Correct. \( a_{12}=a_{21}=6 \).
(D) \( \begin{bmatrix}0 & -5 & 3\\5 & 0 & -7\\-3 & 7 & 0\end{bmatrix} \) is a symmetric matrix — Incorrect. \( a_{12}=-5 \) but \( a_{21}=5 \); the matrix is in fact skew-symmetric.
Shortcut: For (B), Cayley-Hamilton for a 2 × 2 matrix: \( A^2-(\mathrm{tr}\,A)A+|A|I=O \). Here tr A = 4 and |A| = 4 − 4 = 0, so \( A^2=4A \) at once.