Let M be any 3 × 3 matrix with entries from the set \ 0, 1, 2\ . The maximum number of such matrices, for which the sum of diagonal elements of M ^ T M is seven, is______.

Correct answer: 0 [ cases a & b & c d & e & f g & h & i cases ] [ cases a & d & g b & e & h c & f & i cases ] a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 + e ^ 2 + f ^ 2 + g ^ 2 + h ^ 2 + i ^ 2 = 7 Case-I : Seven (1 ' s) and two (0 ' s) C2 9 = 36 Case-II: One (2) and three (1 ' s) and five (0 ' s) (9 !)/(5 ! 3 !) = 504 ∴ Total = 540

JEE Main 202124 FebruaryMorning Shift

Exam: JEE Main
Year: 2021
Date: 24 February
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2021 · 24 February · Morning Shift · Mathematics · Matrices

Let M be any 3 × 3 matrix with entries from the set \ 0, 1, 2\ ....

Let

M

be any

3 \times 3

matrix with entries from the set \left\{0, 1, 2\right\}

. The maximum number of such matrices, for which the sum of diagonal elements of

{M}^{T}M is seven, is______.

Correct answer

Step-by-step solution

\left[

\begin{cases}a & b & c \\ d & e & f \\ g & h & i\end{cases}

\right]

\left[

\begin{cases}a & d & g \\ b & e & h \\ c & f & i\end{cases}

\right]

${a}^{2} + {b}^{2} + {c}^{2} + {d}^{2} + {e}^{2} + {f}^{2} + {g}^{2} + {h}^{2} + {i}^{2} = 7$

Case-I : Seven $\left(1 ' s\right)$ and two $\left(0 ' s\right)$

$C2 9 = 36$

Case-II: One \left(2\right) $and three$ \left(1 ' s\right) $and five \left(0 ' s\right)$

$\dfrac{9 !}{5 ! 3 !} = 504$

$∴$ Total $= 540$

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