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Calculate Expression of Matrix Parameters from Adjoint Relation

Let A=[−11−1101001]A = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix} satisfy
A2+α(adj(adj(A)))+β(adj(A)(adj(adj(A))))=[2−22−20−100−1]A^2 + \alpha(\text{adj}(\text{adj}(A))) + \beta(\text{adj}(A)(\text{adj}(\text{adj}(A)))) = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}
for some α,β∈R\alpha, \beta \in \mathbb{R}. Then (α−β)2(\alpha - \beta)^2 is equal to _______.

Official Numerical Answer4

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MatricesMatricesDeterminants

Step-by-Step Solution

To solve the given matrix equation, we first determine the determinant of AA: det⁡(A)=−1\det(A) = -1

Using standard properties of adjoint matrices for a 3×33 \times 3 matrix, we simplify the terms: adj(adj(A))=det⁡(A)A=−A\text{adj}(\text{adj}(A)) = \det(A) A = -A adj(A)adj(adj(A))=det⁡(A)I=I\text{adj}(A)\text{adj}(\text{adj}(A)) = \det(A) I = I

Substituting these identities into the given equation gives: A2−αA+βI=[2−22−20−100−1]A^2 - \alpha A + \beta I = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}

Calculating A2A^2 directly: A2=[2−11−110001]A^2 = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

By equating corresponding entries on both sides of the matrix equation, we obtain: α=1\alpha = 1 β=−1\beta = -1

Finally, evaluating the required expression: (α−β)2=(1−(−1))2=4(\alpha - \beta)^2 = (1 - (-1))^2 = 4

Calculate Expression of Matrix Parameters from Adjoint Relation | Mathematics PYQ Solution - JEE Challenger