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 A A ^ T is 3, is ___________.

Correct answer: 0 Let, A = [ a _ ii ] _ 3 × 3 tr ( A A ^ T ) = 3 a _ 11 ^ 2 + a _ 12 ^ 2 + a _ 13 ^ 2 + a _ 21 ^ 2 + . . . . . . . + a _ 33 ^ 2 = 3 Possible cases cases 0, 0,0, 0,0, 0,1, 1,1 & → 1 0, 0,0, 0,0, 0, - 1, - 1, - 1 & → 1 0, 0,0, 0,0, 0,1, 1, - 1 & → 3 0, 0,0, 0,0, 0, - 1,1, - 1 & → 3 cases \ ^ 9 C _ 6 × 8 = 84 × 8 = 672

JEE Main 20208 JanuaryMorning Shift

Exam: JEE Main
Year: 2020
Date: 8 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2020 · 8 January · Morning Shift · Mathematics · Matrices

The number of all 3 × 3 matrices A, with entries from the set \ -...

The number of all

3 \times 3

matrices

A,

with entries from the set \left\{- 1, 0, 1\right\}

such that the sum of the diagonal elements of

A{A}^{T} is

3,

is ___________.

Correct answer

Step-by-step solution

Let,

A = {\left[{a}_{ii}\right]}_{3 \times 3}

tr ( A{A}^{T} ) = 3

{a}_{11}^{2} + {a}_{12}^{2} + {a}_{13}^{2} + {a}_{21}^{2} + . . . . . . . + {a}_{33}^{2} = 3

Possible cases
\left\begin{cases}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, 0,1, 1, - 1 & \to 3 \\ 0, 0,0, 0,0, 0, - 1,1, - 1 & \to 3\end{cases}$\right\}{}^{9}{C}_{6} \times 8 = 84 \times 8 = 672

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