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Determinant from the matrix of cofactors (A times adj A = |A| I)

This type of question, asked 2 times: 5 Apr 2024, 29 Jan 2025. All 2 questions below, oldest first.

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  1. Question 1

    5 Apr 2024

    168
    Let αβ≠0\alpha\beta \neq 0 and A=[βα3ααβ−βα2α]A =\begin{bmatrix} \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ - \beta & \alpha & 2\alpha \end{bmatrix}. If B=[3α−93α−α7−2α−2α5−2β]B =\begin{bmatrix} 3\alpha & - 9 & 3\alpha \\ - \alpha & 7 & - 2\alpha \\ - 2\alpha & 5 & - 2\beta \end{bmatrix} is the matrix of cofactors of the elements of AA, then det(AB)det(AB) is equal to :

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  2. Question 2

    29 Jan 2025

    70
    Let A=[aij]=[log⁡5128log⁡45log⁡58log⁡425]A =\left\lbrack a_{ij} \right\rbrack=\begin{bmatrix} \log_{5}128 & \log_{4}5 \\ \log_{5}8 & \log_{4}25 \end{bmatrix}. If AijA_{ij} is the cofactor of aij,Cij=∑k=12aikAjk,1≤ia_{ij},C_{ij}=\sum_{k = 1}^{2} a_{ik}A_{jk},1 \leq i, j≤2j \leq 2, and C=[Cij]C =\left\lbrack C_{ij} \right\rbrack, then 8∣C∣8|C| is equal to :

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