Matrices & Determinants
Skew-Symmetric Matrix
nta_pyq_2025_apr
Grade 12
Question:
Let $A$ be a $3 \times 3$ matrix such that $X^TAX = O$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix}x\\y\\z\end{bmatrix}$. If $A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix}$, $A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix}$, and $\det(\text{adj}(2A + I)) = 2^\alpha 3^\beta 5^\gamma$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is ___
Step-by-Step Solution
Key Concept: $X^TAX = 0$ for all $X$ implies $A$ is skew-symmetric ($a_{ii} = 0$, $a_{ij} = -a_{ji}$). Use the given products to find the entries of $A$, then compute $\det(\text{adj}(2A+I))$.
$A$ is skew-symmetric. From the two given products, solve to get $A = \begin{bmatrix}0&-1&2\\1&0&3\\-2&-3&0\end{bmatrix}$. Then $2(A+I) = \begin{bmatrix}2&-2&4\\2&2&6\\-2&-6&2\end{bmatrix}$, $\det = 120$. $\det(\text{adj}(2A+I)) = 120^2 = 2^6 \cdot 3^2 \cdot 5^2$. $\alpha^2+\beta^2+\gamma^2 = 36+4+4 = 44$.
Correct Answer: 44