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Matrices and Determinants question

2025 · 24 Jan · Shift 1 · Q46

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JEE Main
Mathematics
Matrices and Determinants
2025 · 24 Jan · Shift 1 · Q46

Matrices and Determinants question

2025 · 24 Jan · Shift 1 · Q46

JEE MainMathematicsMatrices and DeterminantsNumerical+4 / −1

Let A be a

3 \times 3

matrix such that

\mathrm{X}^{\mathrm{T}} \mathrm{AX}=\mathrm{O}

for all nonzero

3 \times 1

matrices

X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]

. If

\mathrm{A}\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{c}1 \\ 4 \\ -5\end{array}\right], \mathrm{A}\left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]=\left[\begin{array}{c}0 \\ 4 \\ -8\end{array}\right]

, and

\operatorname{det}(\operatorname{adj}(2(\mathrm{~A}+\mathrm{I})))=2^\alpha 3^\beta 5^\gamma, \alpha, \beta, \gamma \in N

, then

\alpha^2+\beta^2+\gamma^2

is

Numerical answer

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