Let $A$ and $B$ be two square matrices of order $3$ and $AB = O_{3}$, where $O_{3}$ denotes the null matrix of order $3$. Then,

  • A
    must be $A = O_{3}$ and $B = O_{3}$
  • B
    if $A \neq O_{3}$, then $B$ must be $O_{3}$
  • C
    if $A = O_{3}$, then $B$ must be $O_{3}$
  • D
    it is possible that $A \neq O_{3}$ and $B \neq O_{3}$

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The number of all $3 \times 3$ matrices $A$,with entries from the set $\{-1, 0, 1\}$ such that the sum of the diagonal elements of $AA^{T}$ is $3$,is

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