Transpose of matrix

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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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$\begin{array}{l} tr ({AA}^{T}) = 3 \\ a_{11}^{2} + a_{12}^{2} + a_{13}^{2} + a_{21}^{2} + \dots \dots \dots + a_{33}^{2} = 3 ∣ \\ Possible cases \\ \begin{array}{cl} 0, 0, 0, 0, 0, 0, 1, 1, 1 & \to 1 \\ 0, 0, 0, 0, 0, 0, - 1, - 1, - 1 & \to 1 \\ 0, 0, 0, 0, 0, 01, 1, - 1 & \to 3 \\ 0, 0, 0, 0, 0, 0, - 1, 1, - 1 & \to 3 \end{array}\} {}^{9}C_{6} \times 8 = 84 \times 8 = 672 \end{array}$

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Transpose of matrix

Remember concepts with our Masterclasses.

The number of all 3×3 matrices A , with entries from the set {-1,0,1} such that the sum of the diagonal elements of AAT is 3 . is

trAAT=3a112+a122+a132+a212+………+a332=3∣ Possible cases 0,0,0,0,0,0,1,1,1→10,0,0,0,0,0,-1,-1,-1→10,0,0,0,0,01,1,-1 →30,0,0,0,0,0,-1,1,-1→3C6 9×8=84×8=672

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