The number of matrices A, which can be formed using the elements of the set such that the sum of all the diagonal elements of is 5, is [2026]

Question

The number of $3 \times 2$ matrices A, which can be formed using the elements of the set ${- 2, - 1, 0, 1, 2}$ such that the sum of all the diagonal elements of $A^{T} A$ is 5, is ________ . [2026]

Smart Achievers · Accepted Answer

(312)

${(\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix})}_{3 \times 2}$

$A^{T} A = {(\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix})}_{2 \times 3} {(\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix})}_{3 \times 2}$

$= (\begin{matrix} a_{1}^{2} + a_{2}^{2} + a_{3}^{2} & \dots \\ \dots & b_{1}^{2} + b_{2}^{2} + b_{3}^{2} \end{matrix})$

$Tr (A^{T} A) = a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + b_{1}^{2} + b_{2}^{2} + b_{3}^{2} = 5$

${2, 1, 0, 0, 0, 0}$

${2, - 1, 0, 0, 0, 0}$

${- 2, 1, 0, 0, 0, 0}$

${- 2, - 1, 0, 0, 0, 0}$

${1, 1, 1, 1, 1, 0}$

$No. of ways = \frac{6!}{4!} \times 4 + 2 \times \frac{6!}{5!} + 2 \times \frac{6!}{4!} + 2 \times \frac{6!}{3! 2!}$

$= \frac{6!}{3!} + 2 \times 6 \times 2 + 2 \times 15 \times 2 \times \frac{6!}{3!}$

$= 120 + 120 + 12 + 60 = 312$

Solution

Q.
The number of $3 \times 2$ matrices A, which can be formed using the elements of the set ${- 2, - 1, 0, 1, 2}$ such that the sum of all the diagonal elements of $A^{T} A$ is 5, is ________ . [2026]

Want Us to Email you About Special Offers & Updates?

Solution

Q. The number of 3×2 matrices A, which can be formed using the elements of the set {-2,-1,0,1,2} such that the sum of all the diagonal elements of ATA is 5, is ________ . [2026]

Want Us to Email you About Special Offers & Updates?

Q.
The number of $3 \times 2$ matrices A, which can be formed using the elements of the set ${- 2, - 1, 0, 1, 2}$ such that the sum of all the diagonal elements of $A^{T} A$ is 5, is ________ . [2026]