QJEEM25Mab7cd: JEE Main 2025 Mathematics - Class 12 JEEM25Mab7cd Solution: Let A be a matrix such that for all nonzero matrices . If , and …

$\begin{aligned}& X^T A X=0 \\& (x y z)\left(\begin{array}{lll}a_1 & a_2 & a_3 \\b_1 & b_2 & b_3 \\c_1 & c_2 & c_3\end{array}\right)\left(\begin{array}{l}x \\y \\z\end{array}\right)=0 \\& (x y z)\left(\begin{array}{l}a_1 x+a_2 y+a_3 z \\b_1 x+b_2 y+b_3 z \\c_1 x+c_2 y+c_3 z\end{array}\right)=0\end{aligned}$

$\begin{aligned}& x\left(a_1 x+a_2 y+a_3 z\right)+y\left(b_1 x+b_2 y+b_3 z\right) \\& +z\left(c_1 x+c_2 y+c_3 z\right)=0 \\& a_1=0, b_2=0 c_3=0 \\& a_2+b_1=0, a_3+c_1=0, b_3=c_2=0\end{aligned}$

$\begin{aligned}& A=\text { skew symm matrix } \\& A=\left(\begin{array}{ccc}0 & x & y \\-x & 0 & z \\-y & -z & 0\end{array}\right) ; \quad A=\left(\begin{array}{l}1 \\1 \\1\end{array}\right)=\left(\begin{array}{l}1 \\4 \\-5\end{array}\right) \\& \Rightarrow A=\left(\begin{array}{ccc}0 & x & y \\-x & 0 & z \\-y & -z & 0\end{array}\right)\left(\begin{array}{l}1 \\1 \\1\end{array}\right)=\left(\begin{array}{l}1 \\4 \\-5\end{array}\right)\end{aligned}$

$\begin{aligned}& x+y=1 \\& -x+z=4 \\& y+z=5 \\& \left(\begin{array}{ccc}0 & x & y \\-x & 0 & z \\-y & -z & 0\end{array}\right)\left(\begin{array}{l}1 \\2 \\1\end{array}\right)=\left(\begin{array}{l}1 \\4 \\-8\end{array}\right)\end{aligned}$

$\begin{array}{ll}2 x+y=0 & x=-1 \\-x+z=4 & y=2 \\-y-2 z=-8 & z=3\end{array}$

$\begin{aligned}& A=\left(\begin{array}{ccc}0 & -1 & 2 \\1 & 0 & 3 \\-2 & -3 & 0\end{array}\right) \\& 2(\mathrm{~A}+\mathrm{I})=\left(\begin{array}{ccc}2 & -2 & 4 \\2 & 2 & 6 \\-2 & -6 & 2\end{array}\right) \\& 2(\mathrm{~A}+\mathrm{I})=120 \Rightarrow \operatorname{det}|\operatorname{adi}(2(\mathrm{~A}+\mathrm{I}))| \\& =120^2=2^6 \cdot 3^2 \cdot 5^2 \\& \alpha=6, \beta=2, \gamma=2\end{aligned}$