Question

Let A be a 3 x 3 matrix such that $X^T AX = O$ for all nonzero 3 x 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 $det(adj(2(A + 1))) - 2^\alpha 3^\beta 5^\gamma, \alpha, \beta, \gamma \in N$, then $\alpha^2 + \beta^2 + \gamma^2$ is ____.

Answer 1

44

Answer 2

Since $X^TAX=0$ for all $X$, $A$ is skew-symmetric.

Solve using given conditions to find $a=-1$, $b=2$, $c=3$.

Compute $A+I$ and then $\det(2(A+I))=120$.

Use $\det(\operatorname{adj}(M))=(\det M)^2$ to get $14400=2^6\cdot3^2\cdot5^2$.

Finally, $\alpha^2+\beta^2+\gamma^2=36+4+4=44$.