Let A be a square matrix such that a_{ij} in | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

$A$ must be orthogonal

Vedclass · Answer

(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:

Similar Questions

If $A$ and $B$ are square matrices of order $3 \times 3$,$A$ is non-singular,and $AB = O$,then $B$ is a:

If $A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$ and $\alpha, \beta \in \mathbb{R}$ are such that $\alpha A^2 - \beta A = 2I$, then $\alpha^2 + \beta =$

If $A$ and $B$ are two matrices and $(A + B)(A - B) = A^2 - B^2$,then

Let $A=\begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}$. If the sum of the diagonal elements of $A^{13}$ is $3^{n}$,then $n$ is equal to ..........

Vedclass Test Series

Exam Paper Generator

Online Exam Module