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NVAMediumJEE Main 2025 · 24 January, Shift 1Inverse & Adjoint of a Matrix

Mathematics Question from JEE Main 2025 · 24 January, Shift 1

Let AA be a 3×33 \times 3 matrix such that XTAX=OX^T A X = O for all nonzero 3×13 \times 1 matrices X=[xyz]X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}.

Image shows the vector $$X=\begin{pmatrix}x\\y\\z\end{pmatrix}$$ and the given relations $$A\begin{pmatrix}1\\1\\1\end{pmatrix}=\begin{pmatrix}1\\4\\-5\end{pmatrix}$$, $$A\begin{pmatrix}1\\2\\1\end{pmatrix}=\begin{pmatrix}0\\4\\-8\end{pmatrix}$$, followed by the determinant expression involving adjugate of $$2(A+I)$$ and asking for $$\alpha^2+\beta^2+\gamma^2$$.

If

A[111]=[14−5],A[121]=[04−8]A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}, \qquad A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix}

and det⁡(adj⁡(2(A+I)))=2α3β5γ\det\left(\operatorname{adj}\left(2(A + I)\right)\right) = 2^{\alpha} 3^{\beta} 5^{\gamma} with α,β,γ∈N\alpha, \beta, \gamma \in \mathbb{N}, then α2+β2+γ2\alpha^2 + \beta^2 + \gamma^2 is _____.

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