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  4. The positive value of the determinant of the matrix A, whose textAdj( textAdj(A))= beginpmatrix14 28 -14 -14 14 28 28 -14 14 endpmatrix, is.

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Q. The positive value of the determinant of the matrix A, whose $\text{Adj}(\text{Adj}(A))=\begin{pmatrix}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{pmatrix}$, is_____.

JEE MainJEE Main 2022Matrices

Solution:

$\text{Adj}(\text{Adj} A)=\begin{bmatrix}14 & 18 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{bmatrix}$
$|\text{Adj}(\text{Adj} A)|=\begin{bmatrix}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{bmatrix}=14 \times 14 \times 14\begin{vmatrix}1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1\end{vmatrix}$
$=(14)^{3}[3-2(-5)-1(-1)]=(14)^{3}[14]=(14)^{4}$
$|A|^{4}=(14)^{4} \Rightarrow|A|=14$