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(D) square matrix $A$ with exactly one non-zero entry in each row and each column,where the non-zero entry is $\pm 1$,is known as a generalized permut

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$A$ must be orthogonal

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(D) square matrix $A$ with exactly one non-zero entry in each row and each column,where the non-zero entry is $\pm 1$,is known as a generalized permutation matrix.Let $A$ be an $n 	imes n$ matrix. Since each row and column has exactly one non-zero entry equal to $1$ or $-1$,the product $AA^T$ will result in the identity matrix $I_n$.Specifically,the $(i, j)$-th entry of $AA^T$ is the dot product of the $i$-th row and $j$-th row of $A$. If $i 
eq j$,the rows are orthogonal,so the dot product is $0$. If $i = j$,the dot product is $(\pm 1)^2 = 1$.Thus,$AA^T = I_n$,which implies that $A$ is an orthogonal matrix.Since $A$ is orthogonal,$\det(A) = \pm 1$,so $A$ is always non-singular.

Let $A$ be a square matrix such that $a_{ij} \in \{-1, 0, 1\}$ for all $i, j$ and it has exactly one non-zero entry in each row as well as in each column. Then:

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