Let T and U be the set of all orthogonal matr | vedclass

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(B) The set $A = \{-1, 0, 1\}$ is a finite set with cardinality $n(A) = 3$. The set $T$ consists of all orthogonal matrices of order $3 	imes 3$. The

Vedclass · Accepted Answer

There does not exist a bijective mapping between $A$ and $T$, or between $A$ and $U$.

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(B) The set $A = \{-1, 0, 1\}$ is a finite set with cardinality $n(A) = 3$. The set $T$ consists of all orthogonal matrices of order $3 	imes 3$. The group of orthogonal matrices $O(3)$ is an infinite set. The set $U$ consists of all non-singular matrices of order $3 	imes 3$, which is the general linear group $GL(3, \mathbb{R})$. This is also an infinite set. $A$ bijective mapping between two sets exists if and only if they have the same cardinality. Since $n(A) = 3$ and both $T$ and $U$ are infinite sets, $n(A) 
eq n(T)$ and $n(A) 
eq n(U)$. Therefore, there does not exist a bijective mapping between $A$ and $T$, nor between $A$ and $U$.

Let $T$ and $U$ be the set of all orthogonal matrices of order $3$ over $\mathbb{R}$ and the set of all non-singular matrices of order $3$ over $\mathbb{R}$ respectively. Let $A = \{-1, 0, 1\}$. Then:

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