Number of all 3 × 3 matrices A , with enteries from set {-1,0,1} such that sum of diagonal elements

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3 and 4 .Determinants and Matrices

hard

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

A

$632$

B

$672$

C

$682$

D

$638$

(JEE MAIN-2020)

Solution

$\operatorname{trace}\left(\mathrm{AA}^{\mathrm{T}}\right)=\Sigma \mathrm{a}_{\mathrm{ij}}^{2}=3$

Hence, number of such matrices

$=^{9} \mathrm{C}_{3} \times 2^{3}=672.00$

Standard 12

Mathematics

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