Let S be the set containing all 3 × 3 matrices with entries from -1,0,1 . The total number of matrices A ∈ S such that the sum of all the diagonal elements of A T A is 6 is

Question

Let S be the set containing all 3 × 3 matrices with entries from -1,0,1 . The total number of matrices A ∈ S such that the sum of all the diagonal elements of A T A is 6 is  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

operatornameTr(A AT)=6 AA T = [a d g b e h c f i][ a b c d e f g h i] Now give a2+d2+g2+b2+e2+h2+c2+f2+i2=6 = 9 C3 × 26 =5376

Q. Let $S$ be the set containing all $3 \times 3$ matrices with entries from ${- 1, 0, 1}$ . The total number of matrices A $\in S$ such that the sum of all the diagonal elements of $A^{T} A$ is 6 is _____

$Tr (A A^{T}) = 6$
$A A^{T} = ⎣ ⎡ a b c d e f g h i ⎦ ⎤ ⎣ ⎡ a d g b e h c f i ⎦ ⎤$
Now give
$a^{2} + d^{2} + g^{2} + b^{2} + e^{2} + h^{2} + c^{2} + f^{2} + i^{2} = 6$
$=^{9} C_{3} \times 2^{6}$
$= 5376$

Q. Let S be the set containing all 3×3 matrices with entries from {−1,0,1}. The total number of matrices A ∈S such that the sum of all the diagonal elements of ATA is 6 is _____

Tr(AAT)=6 AAT=⎣⎡​abc​def​ghi​⎦⎤​⎣⎡​adg​beh​cfi​⎦⎤​ Now give a2+d2+g2+b2+e2+h2+c2+f2+i2=6 =9C3​×26 =5376

Q. Let $S$ be the set containing all $3 \times 3$ matrices with entries from ${- 1, 0, 1}$ . The total number of matrices A $\in S$ such that the sum of all the diagonal elements of $A^{T} A$ is 6 is _____

$Tr (A A^{T}) = 6$
$A A^{T} = ⎣ ⎡ a b c d e f g h i ⎦ ⎤ ⎣ ⎡ a d g b e h c f i ⎦ ⎤$
Now give
$a^{2} + d^{2} + g^{2} + b^{2} + e^{2} + h^{2} + c^{2} + f^{2} + i^{2} = 6$
$=^{9} C_{3} \times 2^{6}$
$= 5376$